2018
2018
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Paper 1, Section I, E
Part IA, 2018 commentProve that an increasing sequence in that is bounded above converges.
Let be a decreasing function. Let and . Prove that as .
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Paper 1, Section I,
Part IA, 2018 commentDefine the radius of convergence of a complex power series . Prove that converges whenever and diverges whenever .
If for all does it follow that the radius of convergence of is at least that of ? Justify your answer.
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Paper 1, Section II, F
Part IA, 2018 comment(a) Let be a function, and let . Define what it means for to be continuous at . Show that is continuous at if and only if for every sequence with .
(b) Let be a non-constant polynomial. Show that its image is either the real line , the interval for some , or the interval for some .
(c) Let , let be continuous, and assume that holds for all . Show that must be constant.
Is this also true when the condition that be continuous is dropped?
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Paper 1, Section II,
Part IA, 2018 comment(a) Let be differentiable at . Show that is continuous at .
(b) State the Mean Value Theorem. Prove the following inequalities:
and
(c) Determine at which points the following functions from to are differentiable, and find their derivatives at the points at which they are differentiable:
(d) Determine the points at which the following function from to is continuous:
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Paper 1, Section II, E
Part IA, 2018 commentState and prove the Comparison Test for real series.
Assume for all . Show that if converges, then so do and . In each case, does the converse hold? Justify your answers.
Let be a decreasing sequence of positive reals. Show that if converges, then as . Does the converse hold? If converges, must it be the case that as ? Justify your answers.
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Paper 1, Section II, D
Part IA, 2018 comment(a) Let be a fixed enumeration of the rationals in . For positive reals , define a function from to by setting for each and for irrational. Prove that if then is Riemann integrable. If , can be Riemann integrable? Justify your answer.
(b) State and prove the Fundamental Theorem of Calculus.
Let be a differentiable function from to , and set for . Must be Riemann integrable on ?
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Paper 2, Section I, B
Part IA, 2018 commentConsider the following difference equation for real :
where is a real constant.
For find the steady-state solutions, i.e. those with for all , and determine their stability, making it clear how the number of solutions and the stability properties vary with . [You need not consider in detail particular values of which separate intervals with different stability properties.]
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Paper 2, Section I, B
Part IA, 2018 commentShow that for given there is a function such that, for any function ,
if and only if
Now solve the equation
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Paper 2, Section II, B
Part IA, 2018 commentConsider the differential equation
What values of are ordinary points of the differential equation? What values of are singular points of the differential equation, and are they regular singular points or irregular singular points? Give clear definitions of these terms to support your answers.
For not equal to an integer there are two linearly independent power series solutions about . Give the forms of the two power series and the recurrence relations that specify the relation between successive coefficients. Give explicitly the first three terms in each power series.
For equal to an integer explain carefully why the forms you have specified do not give two linearly independent power series solutions. Show that for such values of there is (up to multiplication by a constant) one power series solution, and give the recurrence relation between coefficients. Give explicitly the first three terms.
If is a solution of the above second-order differential equation then
where is an arbitrarily chosen constant, is a second solution that is linearly independent of . For the case , taking to be a power series, explain why the second solution is not a power series.
[You may assume that any power series you use are convergent.]
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Paper 2, Section II, B
Part IA, 2018 commentThe temperature in an oven is controlled by a heater which provides heat at rate . The temperature of a pizza in the oven is . Room temperature is the constant value .
and satisfy the coupled differential equations
where and are positive constants. Briefly explain the various terms appearing in the above equations.
Heating may be provided by a short-lived pulse at , with or by constant heating over a finite period , with , where and are respectively the Dirac delta function and the Heaviside step function. Again briefly, explain how the given formulae for and are consistent with their description and why the total heat supplied by the two heating protocols is the same.
For . Find the solutions for and for , for each of and , denoted respectively by and , and and . Explain clearly any assumptions that you make about continuity of the solutions in time.
Show that the solutions and tend respectively to and in the limit as and explain why.
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Paper 2, Section II, B
Part IA, 2018 commentBy choosing a suitable basis, solve the equation
subject to the initial conditions .
Explain briefly what happens in the cases or .
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Paper 2, Section II, B
Part IA, 2018 commentThe function satisfies the partial differential equation
where and are non-zero constants.
Defining the variables and , where and are constants, and writing show that
where you should determine the functions and .
If the quadratic has distinct real roots then show that and can be chosen such that and .
If the quadratic has a repeated root then show that and can be chosen such that and .
Hence find the general solutions of the equations
and
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Paper 4, Section I, 3A
Part IA, 2018 comment(a) Define an inertial frame.
(b) Specify three different types of Galilean transformation on inertial frames whose space coordinates are and whose time coordinate is .
(c) State the Principle of Galilean Relativity.
(d) Write down the equation of motion for a particle in one dimension in a potential . Prove that energy is conserved. A particle is at position at time . Find an expression for time as a function of in terms of an integral involving .
(e) Write down the values of any equilibria and state (without justification) whether they are stable or unstable for:
(i)
(ii) for and .
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Paper 4, Section I, A
Part IA, 2018 commentExplain what is meant by a central force acting on a particle moving in three dimensions.
Show that the angular momentum of a particle about the origin for a central force is conserved, and hence that its path lies in a plane.
Show that, in the approximation in which the Sun is regarded as fixed and only its gravitational field is considered, a straight line joining the Sun and an orbiting planet sweeps out equal areas in equal time (Kepler's second law).
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Paper 4, Section II, A
Part IA, 2018 commentConsider a rigid body, whose shape and density distribution are rotationally symmetric about a horizontal axis. The body has mass , radius and moment of inertia about its axis of rotational symmetry and is rolling down a non-slip slope inclined at an angle to the horizontal. By considering its energy, calculate the acceleration of the disc down the slope in terms of the quantities introduced above and , the acceleration due to gravity.
(a) A sphere with density proportional to (where is distance to the sphere's centre and is a positive constant) is launched up a non-slip slope of constant incline at the same time, level and speed as a vertical disc of constant density. Find such that the sphere and the disc return to their launch points at the same time.
(b) Two spherical glass marbles of equal radius are released from rest at time on an inclined non-slip slope of constant incline from the same level. The glass in each marble is of constant and equal density, but the second marble has two spherical air bubbles in it whose radii are half the radius of the marbles, initially centred directly above and below the marble's centre, respectively. Each bubble is centred half-way between the centre of the marble and its surface. At a later time , find the ratio of the distance travelled by the first marble and the second. [ You may state without proof any theorems that you use and neglect the mass of air in the bubbles. ]
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Paper 4, Section II, A
Part IA, 2018 commentDefine the 4-momentum of a particle of rest mass and velocity . Calculate the power series expansion of the component for small (where is the speed of light in vacuo) up to and including terms of order , and interpret the first two terms.
(a) At CERN, anti-protons are made by colliding a moving proton with another proton at rest in a fixed target. The collision in question produces three protons and an anti-proton. Assume that the rest mass of a proton is identical to the rest mass of an anti-proton. What is the smallest possible speed of the incoming proton (measured in the laboratory frame)?
(b) A moving particle of rest mass decays into particles with 4 -momenta , and rest masses , where . Show that
Thus, show that
(c) A particle decays into particle and a massless particle 1 . Particle subsequently decays into particle and a massless particle 2 . Show that
where and are the 4-momenta of particles 1 and 2 respectively and are the masses of particles and respectively.
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Paper 4, Section II, A
Part IA, 2018 commentWrite down the Lorentz force law for a charge travelling at velocity in an electric field and magnetic field .
In a space station which is in an inertial frame, an experiment is performed in vacuo where a ball is released from rest a distance from a wall. The ball has charge and at time , it is a distance from the wall. A constant electric field of magnitude points toward the wall in a perpendicular direction, but there is no magnetic field. Find the speed of the ball on its first impact.
Every time the ball bounces, its speed is reduced by a factor . Show that the total distance travelled by the ball before it comes to rest is
where and are quadratic functions which you should find explicitly.
A gas leak fills the apparatus with an atmosphere and the experiment is repeated. The ball now experiences an additional drag force , where is the instantaneous velocity of the ball and . Solve the system before the first bounce, finding an explicit solution for the distance between the ball and the wall as a function of time of the form
where is a function and and are dimensional constants, all of which you should find explicitly.
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Paper 4, Section II, A
Part IA, 2018 commentThe position and velocity of a particle of mass are measured in a frame which rotates at constant angular velocity with respect to an inertial frame. The particle is subject to a force . What is the equation of motion of the particle?
Find the trajectory of the particle in the coordinates if and at and .
Find the maximum value of the speed of the particle and the times at which it travels at this speed.
[Hint: You may find using the variable helpful.]
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Paper 3, Section I, D
Part IA, 2018 commentFind the order and the sign of the permutation .
How many elements of have order And how many have order
What is the greatest order of any element of ?
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Paper 3, Section I, D
Part IA, 2018 commentProve that every member of is a product of at most three reflections.
Is every member of a product of at most two reflections? Justify your answer.
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Paper 3, Section II, D
Part IA, 2018 commentDefine the sign of a permutation . You should show that it is well-defined, and also that it is multiplicative (in other words, that it gives a homomorphism from to .
Show also that (for ) this is the only surjective homomorphism from to .
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Paper 3, Section II, D
Part IA, 2018 commentLet be an element of a group . We define a map from to by sending to . Show that is an automorphism of (that is, an isomorphism from to ).
Now let denote the group of automorphisms of (with the group operation being composition), and define a map from to by setting . Show that is a homomorphism. What is the kernel of ?
Prove that the image of is a normal subgroup of .
Show that if is cyclic then is abelian. If is abelian, must be abelian? Justify your answer.
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Paper 3, Section II, D
Part IA, 2018 commentDefine the quotient group , where is a normal subgroup of a group . You should check that your definition is well-defined. Explain why, for finite, the greatest order of any element of is at most the greatest order of any element of .
Show that a subgroup of a group is normal if and only if there is a homomorphism from to some group whose kernel is .
A group is called metacyclic if it has a cyclic normal subgroup such that is cyclic. Show that every dihedral group is metacyclic.
Which groups of order 8 are metacyclic? Is metacyclic? For which is metacyclic?
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Paper 3, Section II, D
Part IA, 2018 commentState and prove the Direct Product Theorem.
Is the group isomorphic to Is isomorphic to ?
Let denote the group of all invertible complex matrices with , and let be the subgroup of consisting of those matrices with determinant
Determine the centre of .
Write down a surjective homomorphism from to the group of all unit-length complex numbers whose kernel is . Is isomorphic to ?
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Paper 4, Section I, E
Part IA, 2018 commentState Fermat's theorem.
Let be a prime such that . Prove that there is no solution to
Show that there are infinitely many primes congruent to .
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Paper 4, Section I, E
Part IA, 2018 commentGiven , show that is either an integer or irrational.
Let and be irrational numbers and be rational. Which of and must be irrational? Justify your answers. [Hint: For the last part consider .]
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Paper 4, Section II,
Part IA, 2018 commentLet be a positive integer. Show that for any coprime to , there is a unique such that . Show also that if and are integers coprime to , then is also coprime to . [Any version of Bezout's theorem may be used without proof provided it is clearly stated.]
State and prove Wilson's theorem.
Let be a positive integer and be a prime. Show that the exponent of in the prime factorisation of ! is given by where denotes the integer part of .
Evaluate and
Let be a prime and . Let be the exponent of in the prime factorisation of . Find the exponent of in the prime factorisation of , in terms of and .
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Paper 4, Section II, E
Part IA, 2018 commentFor let denote the set of all sequences of length . We define the distance between two elements and of to be the number of coordinates in which they differ. Show that for all .
For and let . Show that .
A subset of is called a -code if for all with . Let be the maximum possible value of for a -code in . Show that
Find , carefully justifying your answer.
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Paper 4, Section II,
Part IA, 2018 commentLet and be subsets of a finite set . Let . Show that if belongs to for exactly values of , then
where with the convention that , and denotes the indicator function of Hint: Set and consider for which one has .]
Use this to show that the number of elements of that belong to for exactly values of is
Deduce the Inclusion-Exclusion Principle.
Using the Inclusion-Exclusion Principle, prove a formula for the Euler totient function in terms of the distinct prime factors of .
A Carmichael number is a composite number such that for every integer coprime to . Show that if is the product of distinct primes satisfying for , then is a Carmichael number.
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Paper 4, Section II, 8E
Part IA, 2018 commentDefine what it means for a set to be countable.
Show that for any set , there is no surjection from onto the power set . Deduce that the set of all infinite sequences is uncountable.
Let be the set of sequences of subsets of such that for all and . Let consist of all members of for which for all but finitely many . Let consist of all members of for which for all but finitely many . For each of and determine whether it is countable or uncountable. Justify your answers.
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Paper 2, Section I, F
Part IA, 2018 commentLet and be independent Poisson random variables with parameters and respectively.
(i) Show that is Poisson with parameter .
(ii) Show that the conditional distribution of given is binomial, and find its parameters.
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Paper 2, Section I, F
Part IA, 2018 comment(a) State the Cauchy-Schwarz inequality and Markov's inequality. State and prove Jensen's inequality.
(b) For a discrete random variable , show that implies that is constant, i.e. there is such that .
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Paper 2, Section II, F
Part IA, 2018 comment(a) Let and be independent discrete random variables taking values in sets and respectively, and let be a function.
Let . Show that
Let . Show that
(b) Let be independent Bernoulli random variables. For any function , show that
Let denote the set of all sequences of length . By induction, or otherwise, show that for any function ,
where and .
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Paper 2, Section II, 10F
Part IA, 2018 comment(a) Let and be independent random variables taking values , each with probability , and let . Show that and are pairwise independent. Are they independent?
(b) Let and be discrete random variables with mean 0 , variance 1 , covariance . Show that .
(c) Let be discrete random variables. Writing , show that .
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Paper 2, Section II, F
Part IA, 2018 comment(a) Consider a Galton-Watson process . Prove that the extinction probability is the smallest non-negative solution of the equation where . [You should prove any properties of Galton-Watson processes that you use.]
In the case of a Galton-Watson process with
find the mean population size and compute the extinction probability.
(b) For each , let be a random variable with distribution . Show that
in distribution, where is a standard normal random variable.
Deduce that
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Paper 2, Section II, F
Part IA, 2018 commentFor a symmetric simple random walk on starting at 0 , let .
(i) For and , show that
(ii) For , show that and that
(iii) Prove that .
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Paper 3, Section I, C
Part IA, 2018 commentDerive a formula for the curvature of the two-dimensional curve .
Verify your result for the semicircle with radius given by .
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Paper 3, Section I,
Part IA, 2018 commentIn plane polar coordinates , the orthonormal basis vectors and satisfy
Hence derive the expression for the Laplacian operator .
Calculate the Laplacian of , where and are constants. Hence find all solutions to the equation
Explain briefly how you know that there are no other solutions.
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Paper 3, Section II, C
Part IA, 2018 commentGiven a one-to-one mapping and between the region in the -plane and the region in the -plane, state the formula for transforming the integral into an integral over , with the Jacobian expressed explicitly in terms of the partial derivatives of and .
Let be the region and consider the change of variables and . Sketch , the curves of constant and the curves of constant in the -plane. Find and sketch the image of in the -plane.
Calculate using this change of variables. Check your answer by calculating directly.
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Paper 3, Section II, C
Part IA, 2018 commentState the formula of Stokes's theorem, specifying any orientation where needed.
Let . Calculate and verify that .
Sketch the surface defined as the union of the surface and the surface .
Verify Stokes's theorem for on .
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Paper 3, Section II, C
Part IA, 2018 commentUse Maxwell's equations,
to derive expressions for and in terms of and .
Now suppose that there exists a scalar potential such that , and as . If is spherically symmetric, calculate using Gauss's flux method, i.e. by integrating a suitable equation inside a sphere centred at the origin. Use your result to find and in the case when for and otherwise.
For each integer , let be the sphere of radius centred at the point . Suppose that vanishes outside , and has the constant value in the volume between and for . Calculate and at the point .
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Paper 3, Section II, C
Part IA, 2018 comment(a) Suppose that a tensor can be decomposed as
where is symmetric. Obtain expressions for and in terms of , and check that is satisfied.
(b) State the most general form of an isotropic tensor of rank for , and verify that your answers are isotropic.
(c) The general form of an isotropic tensor of rank 4 is
Suppose that and satisfy the linear relationship , where is isotropic. Express in terms of , assuming that and . If instead and , find all such that .
(d) Suppose that and satisfy the quadratic relationship , where is an isotropic tensor of rank 6 . If is symmetric and is antisymmetric, find the most general non-zero form of and prove that there are only two independent terms. [Hint: You do not need to use the general form of an isotropic tensor of rank 6.]
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Paper 1, Section I, C
Part IA, 2018 commentFor define the principal value of . State de Moivre's theorem.
Hence solve the equations (i) , (ii) , (iii) (iv)
[In each expression, the principal value is to be taken.]
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Paper 1, Section I, A
Part IA, 2018 commentThe map is defined for , where is a unit vector in and is a real constant.
(i) Find the values of for which the inverse map exists, as well as the inverse map itself in these cases.
(ii) When is not invertible, find its image and kernel. What is the value of the rank and the value of the nullity of ?
(iii) Let . Find the components of the matrix such that . When is invertible, find the components of the matrix such that .
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Paper 1, Section II,
Part IA, 2018 commentLet be non-zero real vectors. Define the inner product in terms of the components and , and define the norm . Prove that . When does equality hold? Express the angle between and in terms of their inner product.
Use suffix notation to expand .
Let be given unit vectors in , and let . Obtain expressions for the angle between and each of and , in terms of and . Calculate for the particular case when the angles between and are all equal to , and check your result for an example with and an example with .
Consider three planes in passing through the points and , respectively, with unit normals and , respectively. State a condition that must be satisfied for the three planes to intersect at a single point, and find the intersection point.
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Paper 1, Section II, B
Part IA, 2018 comment(a) Consider the matrix
representing a rotation about the -axis through an angle .
Show that has three eigenvalues in each with modulus 1 , of which one is real and two are complex (in general), and give the relation of the real eigenvector and the two complex eigenvalues to the properties of the rotation.
Now consider the rotation composed of a rotation by angle about the -axis followed by a rotation by angle about the -axis. Determine the rotation axis and the magnitude of the angle of rotation .
(b) A surface in is given by
By considering a suitable eigenvalue problem, show that the surface is an ellipsoid, find the lengths of its semi-axes and find the position of the two points on the surface that are closest to the origin.
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Paper 1, Section II, B
Part IA, 2018 commentLet be a real symmetric matrix.
(a) Prove the following:
(i) Each eigenvalue of is real and there is a corresponding real eigenvector.
(ii) Eigenvectors corresponding to different eigenvalues are orthogonal.
(iii) If there are distinct eigenvalues then the matrix is diagonalisable.
Assuming that has distinct eigenvalues, explain briefly how to choose (up to an arbitrary scalar factor) the vector such that is maximised.
(b) A scalar and a non-zero vector such that
are called, for a specified matrix , respectively a generalised eigenvalue and a generalised eigenvector of .
Assume the matrix is real, symmetric and positive definite (i.e. for all non-zero complex vectors ).
Prove the following:
(i) If is a generalised eigenvalue of then it is a root of .
(ii) Each generalised eigenvalue of is real and there is a corresponding real generalised eigenvector.
(iii) Two generalised eigenvectors , corresponding to different generalised eigenvalues, are orthogonal in the sense that .
(c) Find, up to an arbitrary scalar factor, the vector such that the value of is maximised, and the corresponding value of , where
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Paper 1, Section II, A
Part IA, 2018 commentWhat is the definition of an orthogonal matrix ?
Write down a matrix representing the rotation of a 2-dimensional vector by an angle around the origin. Show that is indeed orthogonal.
Take a matrix
where are real. Suppose that the matrix is diagonal. Determine all possible values of .
Show that the diagonal entries of are the eigenvalues of and express them in terms of the determinant and trace of .
Using the above results, or otherwise, find the elements of the matrix
as a function of , where is a natural number.
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Paper 3, Section I,
Part IB, 2018 commentFor a continuous function , define
Show that
for every continuous function , where denotes the Euclidean norm on .
Find all continuous functions with the property that
regardless of the norm on .
[Hint: start by analysing the case when is the Euclidean norm .]
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Paper 2, Section I, F
Part IB, 2018 commentShow that defines a norm on the space of continuous functions .
Let be the set of continuous functions with . Show that for each continuous function , there is a sequence with such that as
Show that if is continuous and for every then .
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Paper 4, Section I, F
Part IB, 2018 commentState the Bolzano-Weierstrass theorem in . Use it to deduce the BolzanoWeierstrass theorem in .
Let be a closed, bounded subset of , and let be a function. Let be the set of points in where is discontinuous. For and , let denote the ball . Prove that for every , there exists such that whenever and .
(If you use the fact that a continuous function on a compact metric space is uniformly continuous, you must prove it.)
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Paper 1, Section II, F
Part IB, 2018 commentLet be a non-empty open set and let .
(a) What does it mean to say that is differentiable? What does it mean to say that is a function?
If is differentiable, show that is continuous.
State the inverse function theorem.
(b) Suppose that is convex, is and that its derivative at a satisfies for all , where is the identity map and denotes the operator norm. Show that is injective.
Explain why is an open subset of .
Must it be true that ? What if ? Give proofs or counter-examples as appropriate.
(c) Find the largest set such that the map given by satisfies for every .
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Paper 4, Section II, F
Part IB, 2018 comment(a) Define what it means for a metric space to be complete. Give a metric on the interval such that is complete and such that a subset of is open with respect to if and only if it is open with respect to the Euclidean metric on . Be sure to prove that has the required properties.
(b) Let be a complete metric space.
(i) If , show that taken with the subspace metric is complete if and only if is closed in .
(ii) Let and suppose that there is a number such that for every . Show that there is a unique point such that .
Deduce that if is a sequence of points in converging to a point , then there are integers and such that for every .
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Paper 3, Section II, F
Part IB, 2018 comment(a) Let and let be functions for What does it mean to say that the sequence converges uniformly to on ? What does it mean to say that is uniformly continuous?
(b) Let be a uniformly continuous function. Determine whether each of the following statements is true or false. Give reasons for your answers.
(i) If for each and each , then uniformly on .
(ii) If for each and each , then uniformly on .
(c) Let be a closed, bounded subset of . For each , let be a continuous function such that is a decreasing sequence for each . If is such that for each there is with , show that there is such that .
Deduce the following: If is a continuous function for each such that is a decreasing sequence for each , and if the pointwise limit of is a continuous function , then uniformly on .
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Paper 2, Section II, F
Part IB, 2018 comment(a) Let be a metric space, a non-empty subset of and . Define what it means for to be Lipschitz. If is Lipschitz with Lipschitz constant and if
for each , show that for each and that is Lipschitz with Lipschitz constant . (Be sure to justify that , i.e. that the infimum is finite for every .)
(b) What does it mean to say that two norms on a vector space are Lipschitz equivalent?
Let be an -dimensional real vector space equipped with a norm . Let be a basis for . Show that the map defined by is continuous. Deduce that any two norms on are Lipschitz equivalent.
(c) Prove that for each positive integer and each , there is a constant with the following property: for every polynomial of degree , there is a point such that
where is the derivative of .
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Paper 4, Section I, F
Part IB, 2018 comment(a) Let be open, and suppose that . Let be analytic.
State the Cauchy integral formula expressing as a contour integral over . Give, without proof, a similar expression for .
If additionally and is bounded, deduce that must be constant.
(b) If is analytic where are real, and if for all , show that is constant.
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Paper 3, Section II, F
Part IB, 2018 commentLet and let be analytic.
(a) If there is a point such that for all , prove that is constant.
(b) If and for all , prove that for all .
(c) Show that there is a constant independent of such that if and for all then whenever
[Hint: you may find it useful to consider the principal branch of the map .]
(d) Does the conclusion in (c) hold if we replace the hypothesis for with the hypothesis for , and keep all other hypotheses? Justify your answer.
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Paper 1, Section I, A
Part IB, 2018 comment(a) Show that
is a conformal mapping from the right half -plane, , to the strip
for a suitably chosen branch of that you should specify.
(b) Show that
is a conformal mapping from the right half -plane, , to the unit disc
(c) Deduce a conformal mapping from the strip to the disc .
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Paper 1, Section II, A
Part IB, 2018 comment(a) Let be a rectangular contour with vertices at and for some taken in the anticlockwise direction. By considering
show that
(b) By using a semi-circular contour in the upper half plane, calculate
for .
[You may use Jordan's Lemma without proof.]
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Paper 2, Section II, A
Part IB, 2018 comment(a) Let be a complex function. Define the Laurent series of about , and give suitable formulae in terms of integrals for calculating the coefficients of the series.
(b) Calculate, by any means, the first 3 terms in the Laurent series about for
Indicate the range of values of for which your series is valid.
(c) Let
Classify the singularities of for .
(d) By considering
where for some suitably chosen , show that
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Paper 3, Section I, A
Part IB, 2018 comment(a) Let . Define the branch cut of as such that
Show that is an odd function.
(b) Let .
(i) Show that is a branch point of .
(ii) Define the branch cuts of as such that
Find , where denotes just above the branch cut, and denotes just below the branch cut.
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Paper 4, Section II, A
Part IB, 2018 comment(a) Find the Laplace transform of
for .
[You may use without proof that
(b) By using the Laplace transform, show that the solution to
can be written as
for some to be determined.
[You may use without proof that a particular solution to
is given by
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Paper 2, Section I,
Part IB, 2018 commentDerive the Biot-Savart law
from Maxwell's equations, where the time-independent current vanishes outside . [You may assume that the vector potential can be chosen to be divergence-free.]
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Paper 4, Section I,
Part IB, 2018 commentShow that Maxwell's equations imply the conservation of charge.
A conducting medium has where is a constant. Show that any charge density decays exponentially in time, at a rate to be determined.
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Paper 1, Section II, C
Part IB, 2018 commentStarting from the Lorentz force law acting on a current distribution obeying , show that the energy of a magnetic dipole in the presence of a time independent magnetic field is
State clearly any approximations you make.
[You may use without proof the fact that
for any constant vector , and the identity
which holds when is constant.]
A beam of slowly moving, randomly oriented magnetic dipoles enters a region where the magnetic field is
with and constants. By considering their energy, briefly describe what happens to those dipoles that are parallel to, and those that are anti-parallel to the direction of .
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Paper 3, Section II, C
Part IB, 2018 commentUse Maxwell's equations to show that
where is a bounded region, its boundary and its outward-pointing normal. Give an interpretation for each of the terms in this equation.
A certain capacitor consists of two conducting, circular discs, each of large area , separated by a small distance along their common axis. Initially, the plates carry charges and . At time the plates are connected by a resistive wire, causing the charge on the plates to decay slowly as for some constant . Construct the Poynting vector and show that energy flows radially out of the capacitor as it discharges.
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Paper 2, Section II, C
Part IB, 2018 commentA plane with unit normal supports a charge density and a current density that are each time-independent. Show that the tangential components of the electric field and the normal component of the magnetic field are continuous across the plane.
Albert moves with constant velocity relative to the plane. Find the boundary conditions at the plane on the normal component of the magnetic field and the tangential components of the electric field as seen in Albert's frame.
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Paper 1, Section I, D
Part IB, 2018 commentShow that the flow with velocity potential
in two-dimensional, plane-polar coordinates is incompressible in . Determine the flux of fluid across a closed contour that encloses the origin. What does this flow represent?
Show that the flow with velocity potential
has no normal flow across the line . What fluid flow does this represent in the unbounded plane? What flow does it represent for fluid occupying the domain ?
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Paper 2, Section I, D
Part IB, 2018 commentThe Euler equations for steady fluid flow in a rapidly rotating system can be written
where is the density of the fluid, is its pressure, is the acceleration due to gravity and is the constant Coriolis parameter in a Cartesian frame of reference , with pointing vertically upwards.
Fluid occupies a layer of slowly-varying height . Given that the pressure is constant at and that the flow is approximately horizontal with components , show that the contours of are streamlines of the horizontal flow. What is the leading-order horizontal volume flux of fluid between two locations at which and , where ?
Identify the dimensions of all the quantities involved in your expression for the volume flux and show that your expression is dimensionally consistent.
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Paper 1, Section II, D
Part IB, 2018 commentA layer of fluid of dynamic viscosity , density and uniform thickness flows down a rigid vertical plane. The adjacent air has uniform pressure and exerts a tangential stress on the fluid that is proportional to the surface velocity and opposes the flow, with constant of proportionality . The acceleration due to gravity is .
(a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.
(b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields in terms of the parameters given above.
(c) Show that the surface velocity of the fluid layer is .
(d) Determine the volume flux per unit width of the plane for general values of and its limiting values when and .
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Paper 4, Section II, D
Part IB, 2018 commentA deep layer of inviscid fluid is initially confined to the region , in Cartesian coordinates, with directed vertically upwards. An irrotational disturbance is caused to the fluid so that its upper surface takes position . Determine the linear normal modes of the system and the dispersion relation between the frequencies of the normal modes and their wavenumbers.
If the interface is initially displaced to position and released from rest, where is a small constant, determine its position for subsequent times. How far below the surface will the velocity have decayed to times its surface value?
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Paper 3, Section II, D
Part IB, 2018 commentA soap bubble of radius is attached to the end of a long, narrow straw of internal radius and length , the other end of which is open to the atmosphere. The pressure difference between the inside and outside of the bubble is , where is the surface tension of the soap bubble. At time and the air in the straw is at rest. Assume that the flow of air through the straw is irrotational and consider the pressure drop along the straw to show that subsequently
where is the density of air.
By multiplying the equation by and integrating, or otherwise, determine an implicit equation for and show that the bubble disappears in a time
[Hint: The substitution can be used.]
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Paper 1, Section I, G
Part IB, 2018 comment(a) State the Gauss-Bonnet theorem for spherical triangles.
(b) Prove that any geodesic triangulation of the sphere has Euler number equal to
(c) Prove that there is no geodesic triangulation of the sphere in which every vertex is adjacent to exactly 6 triangles.
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Paper 3, Section I, G
Part IB, 2018 commentConsider a quadrilateral in the hyperbolic plane whose sides are hyperbolic line segments. Suppose angles and are right-angles. Prove that is longer than .
[You may use without proof the distance formula in the upper-half-plane model
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Paper 3, Section II, G
Part IB, 2018 commentLet be an open subset of the plane , and let be a smooth parametrization of a surface . A coordinate curve is an arc either of the form
for some constant and , or of the form
for some constant and . A coordinate rectangle is a rectangle in whose sides are coordinate curves.
Prove that all coordinate rectangles in have opposite sides of the same length if and only if at all points of , where and are the usual components of the first fundamental form, and are coordinates in .
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Paper 2, Section II, G
Part IB, 2018 commentFor any matrix
the corresponding Möbius transformation is
which acts on the upper half-plane , equipped with the hyperbolic metric .
(a) Assuming that , prove that is conjugate in to a diagonal matrix . Determine the relationship between and .
(b) For a diagonal matrix with , prove that
for all not on the imaginary axis.
(c) Assume now that . Prove that fixes a point in .
(d) Give an example of a matrix in that does not preserve any point or hyperbolic line in . Justify your answer.
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Paper 4, Section II, G
Part IB, 2018 commentA Möbius strip in is parametrized by
for , where . Show that the Gaussian curvature is
at
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Paper 3, Section I,
Part IB, 2018 comment(a) Find all integer solutions to .
(b) Find all the irreducibles in of norm 9 .
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Paper 4, Section I, G
Part IB, 2018 comment(a) Show that every automorphism of the dihedral group is equal to conjugation by an element of ; that is, there is an such that
for all .
(b) Give an example of a non-abelian group with an automorphism which is not equal to conjugation by an element of .
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Paper 2, Section ,
Part IB, 2018 commentLet be a principal ideal domain and a non-zero element of . We define a new as follows. We define an equivalence relation on by
if and only if . The underlying set of is the set of -equivalence classes. We define addition on by
and multiplication by .
(a) Show that is a well defined ring.
(b) Prove that is a principal ideal domain.
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Paper 1, Section II, G
Part IB, 2018 comment(a) State Sylow's theorems.
(b) Prove Sylow's first theorem.
(c) Let be a group of order 12. Prove that either has a unique Sylow 3-subgroup or .
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Paper 4, Section II, G
Part IB, 2018 comment(a) State the classification theorem for finitely generated modules over a Euclidean domain.
(b) Deduce the existence of the rational canonical form for an matrix over a field .
(c) Compute the rational canonical form of the matrix
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Paper 3, Section II, G
Part IB, 2018 comment(a) State Gauss's Lemma.
(b) State and prove Eisenstein's criterion for the irreducibility of a polynomial.
(c) Determine whether or not the polynomial
is irreducible over .
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Paper 2, Section II, G
Part IB, 2018 comment(a) Prove that every principal ideal domain is a unique factorization domain.
(b) Consider the ring .
(i) What are the units in ?
(ii) Let be irreducible. Prove that either , for a prime, or and .
(iii) Prove that is not expressible as a product of irreducibles.
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Paper 1, Section I, E
Part IB, 2018 commentState the Rank-Nullity Theorem.
If and are linear maps and is finite dimensional, show that
If is another linear map, show that
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Paper 2, Section I, E
Part IB, 2018 commentLet be a real vector space. Define the dual vector space of . If is a subspace of , define the annihilator of . If is a basis for , define its dual and prove that it is a basis for .
If has basis and is the subspace spanned by
give a basis for in terms of the dual basis .
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Paper 4, Section I, E
Part IB, 2018 commentDefine a quadratic form on a finite dimensional real vector space. What does it mean for a quadratic form to be positive definite?
Find a basis with respect to which the quadratic form
is diagonal. Is this quadratic form positive definite?
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Paper 1, Section II, E
Part IB, 2018 commentDefine a Jordan block . What does it mean for a complex matrix to be in Jordan normal form?
If is a matrix in Jordan normal form for an endomorphism , prove that
is the number of Jordan blocks of with .
Find a matrix in Jordan normal form for . [Consider all possible values of .]
Find a matrix in Jordan normal form for the complex matrix
assuming it is invertible.
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Paper 2, Section II, E
Part IB, 2018 commentIf is an matrix over a field, show that there are invertible matrices and such that
for some , where is the identity matrix of dimension .
For a square matrix of the form with and square matrices, prove that .
If and have no common eigenvalue, show that the linear map
is injective.
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Paper 4, Section II, E
Part IB, 2018 commentLet be a finite dimensional inner-product space over . What does it mean to say that an endomorphism of is self-adjoint? Prove that a self-adjoint endomorphism has real eigenvalues and may be diagonalised.
An endomorphism is called positive definite if it is self-adjoint and satisfies for all non-zero ; it is called negative definite if is positive definite. Characterise the property of being positive definite in terms of eigenvalues, and show that the sum of two positive definite endomorphisms is positive definite.
Show that a self-adjoint endomorphism has all eigenvalues in the interval if and only if is positive definite for all and negative definite for all .
Let be self-adjoint endomorphisms whose eigenvalues lie in the intervals and respectively. Show that all of the eigenvalues of lie in the interval .
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Paper 3, Section II, E
Part IB, 2018 commentState and prove the Cayley-Hamilton Theorem.
Let be an complex matrix. Using division of polynomials, show that if is a polynomial then there is another polynomial of degree at most such that for each eigenvalue of and such that .
Hence compute the entry of the matrix when
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Paper 3, Section I, H
Part IB, 2018 commentThe mathematics course at the University of Barchester is a three-year one. After the end-of-year examinations there are three possibilities:
(i) failing and leaving (probability );
(ii) taking that year again (probability );
(iii) going on to the next year (or graduating, if the current year is the third one) (probability ).
Thus there are five states for a student year, year, year, left without a degree, graduated).
Write down the transition matrix. Classify the states, assuming . Find the probability that a student will eventually graduate.
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Paper 4, Section I, H
Part IB, 2018 commentLet be the transition matrix for an irreducible Markov chain on the finite state space .
(a) What does it mean to say that a distribution is the invariant distribution for the chain?
(b) What does it mean to say that the chain is in detailed balance with respect to a distribution ? Show that if the chain is in detailed balance with respect to a distribution then is the invariant distribution for the chain.
(c) A symmetric random walk on a connected finite graph is the Markov chain whose state space is the set of vertices of the graph and whose transition probabilities are
where is the number of vertices adjacent to vertex . Show that the random walk is in detailed balance with respect to its invariant distribution.
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Paper 1, Section II, H
Part IB, 2018 commentA coin-tossing game is played by two players, and . Each player has a coin and the probability that the coin tossed by player comes up heads is , where . The players toss their coins according to the following scheme: tosses first and then after each head, pays one pound and has the next toss, while after each tail, pays one pound and has the next toss.
Define a Markov chain to describe the state of the game. Find the probability that the game ever returns to a state where neither player has lost money.
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Paper 2, Section II, H
Part IB, 2018 commentFor a finite irreducible Markov chain, what is the relationship between the invariant probability distribution and the mean recurrence times of states?
A particle moves on the vertices of the hypercube, , in the following way: at each step the particle is equally likely to move to each of the adjacent vertices, independently of its past motion. (Two vertices are adjacent if the Euclidean distance between them is one.) The initial vertex occupied by the particle is . Calculate the expected number of steps until the particle
(i) first returns to ,
(ii) first visits ,
(iii) first visits .
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Paper 2, Section I,
Part IB, 2018 commentShow that
along a characteristic curve of the -order pde
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Paper 4, Section I, A
Part IB, 2018 commentBy using separation of variables, solve Laplace's equation
subject to
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Paper 3, Section I, A
Part IB, 2018 comment(a) Determine the Green's function satisfying
with . Here ' denotes differentiation with respect to .
(b) Using the Green's function, solve
with .
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Paper 1, Section II, 14C
Part IB, 2018 commentDefine the convolution of two functions and . Defining the Fourier transform of by
show that
Given that the Fourier transform of is
find the Fourier transform of .
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Paper 3, Section II, A
Part IB, 2018 commentConsider the Dirac delta function, , defined by the sampling property
for any suitable function and real constant .
(a) Show that for any non-zero .
(b) Show that , where denotes differentiation with respect to .
(c) Calculate
where is the derivative of the delta function.
(d) For
show that .
(e) Find expressions in terms of the delta function and its derivatives for
(i)
(ii)
(f) Hence deduce that
[You may assume that
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Paper 2, Section II, A
Part IB, 2018 comment(a) Let be a -periodic function (i.e. for all ) defined on by
Find the Fourier series of in the form
(b) Find the general solution to
where is as given in part (a) and is -periodic.
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Paper 4, Section II, 17C
Part IB, 2018 commentLet be a bounded region in the plane, with smooth boundary . Green's second identity states that for any smooth functions on
where is the outward pointing normal to . Using this identity with replaced by
and taking care of the singular point , show that if solves the Poisson equation then
at any , where all derivatives are taken with respect to .
In the case that is the unit disc , use the method of images to show that the solution to Laplace's equation inside , subject to the boundary condition
is
where are polar coordinates in the disc and is a constant.
[Hint: The image of a point is the point , and then
for all
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Paper 3, Section I,
Part IB, 2018 commentWhat does it mean to say that a topological space is connected? If is a topological space and , show that there is a connected subspace of so that if is any other connected subspace containing then .
Show that the sets partition .
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Paper 2, Section I, E
Part IB, 2018 commentWhat does it mean to say that is a metric on a set ? What does it mean to say that a subset of is open with respect to the metric ? Show that the collection of subsets of that are open with respect to satisfies the axioms of a topology.
For , the set of continuous functions , show that the metrics
give different topologies.
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Paper 1, Section II, E
Part IB, 2018 commentWhat does it mean to say that a topological space is compact? Prove directly from the definition that is compact. Hence show that the unit circle is compact, proving any results that you use. [You may use without proof the continuity of standard functions.]
The set has a topology for which the closed sets are the empty set and the finite unions of vector subspaces. Let denote the set with the subspace topology induced by . By considering the subspace topology on , or otherwise, show that is compact.
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Paper 4, Section II, E
Part IB, 2018 commentLet and for each let
Prove that the set of unions of the sets forms a topology on .
Prove or disprove each of the following:
(i) is Hausdorff;
(ii) is compact.
If and are topological spaces, is the union of closed subspaces and , and is a function such that both and are continuous, show that is continuous. Hence show that is path-connected.
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Paper 1, Section I, D
Part IB, 2018 commentThe Trapezoidal Rule for solving the differential equation
is defined by
where .
Determine the minimum order of convergence of this rule for general functions that are sufficiently differentiable. Show with an explicit example that there is a function for which the local truncation error is for some constant .
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Paper 4 , Section I, D
Part IB, 2018 commentwhere and are real parameters. Find the factorisation of the matrix . For what values of does the equation have a unique solution for ?
For , use the decomposition with forward and backward substitution to determine a value for for which a solution to exists. Find the most general solution to the equation in this case.
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Paper 1, Section II, D
Part IB, 2018 commentShow that if then the matrix transformation
is orthogonal. Show further that, for any two vectors of equal length,
Explain how to use such transformations to convert an matrix with into the form , where is an orthogonal matrix and is an upper-triangular matrix, and illustrate the method using the matrix
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Paper 3, Section II, D
Part IB, 2018 commentTaylor's theorem for functions is given in the form
Use integration by parts to show that
Let be a linear functional on such that for . Show that
where the Peano kernel function You may assume that the functional commutes with integration over a fixed interval.]
The error in the mid-point rule for numerical quadrature on is given by
Show that if is a linear polynomial. Find the Peano kernel function corresponding to explicitly and verify the formula ( ) in the case .
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Paper 2, Section II, D
Part IB, 2018 commentShow that the recurrence relation
where is an inner product on real polynomials, produces a sequence of orthogonal, monic, real polynomials of degree exactly of the real variable , provided that is a monic, real polynomial of degree exactly .
Show that the choice leads to a three-term recurrence relation of the form
where and are constants that should be determined in terms of the inner products and .
Use this recurrence relation to find the first four monic Legendre polynomials, which correspond to the inner product defined by
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Paper 1, Section I, 8H
Part IB, 2018 commentWhat is meant by a transportation problem? Illustrate the transportation algorithm by solving the problem with three sources and three destinations described by the table

where the figures in the boxes denote transportation costs, the right-hand column denotes supplies, and the bottom row denotes requirements.
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Paper 2, Section , H
Part IB, 2018 commentWhat does it mean to state that is a convex function?
Suppose that are convex functions, and for let
Assuming is finite for all , prove that the function is convex.
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Paper 4, Section II, H
Part IB, 2018 commentGiven a network with a source , a sink , and capacities on directed edges, define a cut. What is meant by the capacity of a cut? State the max-flow min-cut theorem. If the capacities of edges are integral, what can be said about the maximum flow?
Consider an matrix in which each entry is either 0 or 1 . We say that a set of lines (rows or columns of the matrix) covers the matrix if each 1 belongs to some line of the set. We say that a set of 1 's is independent if no pair of 1 's of the set lie in the same line. Use the max-flow min-cut theorem to show that the maximal number of independent 1's equals the minimum number of lines that cover the matrix.
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Paper 3, Section II, 21H
Part IB, 2018 commentState and prove the Lagrangian Sufficiency Theorem.
The manufacturers, and , of two competing soap powders must plan how to allocate their advertising resources ( and pounds respectively) among distinct geographical regions. If and denote, respectively, the resources allocated to area by and then the number of packets sold by and in area are
respectively, where is the total market in area , and are known constants. The difference between the amount sold by and is then
seeks to maximize this quantity, while seeks to minimize it.
(i) If knows 's allocation, how should choose ?
(ii) Determine the best strategies for and if each assumes the other will know its strategy and react optimally.
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Paper 4, Section I, B
Part IB, 2018 commentA particle moving in one space dimension with wavefunction obeys the timedependent Schrödinger equation. Write down the probability density and current density in terms of the wavefunction and show that they obey the equation
Evaluate in the case that
where , and and are constants, which may be complex.
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Paper 3, Section I, B
Part IB, 2018 commentWhat is meant by the statement that an operator is Hermitian?
Consider a particle of mass in a real potential in one dimension. Show that the Hamiltonian of the system is Hermitian.
Starting from the time-dependent Schrödinger equation, show that
where is the momentum operator and denotes the expectation value of the operator .
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Paper 1, Section II, B
Part IB, 2018 commentThe relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass under the influence of the central potential
where and are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and a neutron, giving the condition on for this state to exist.
[If is spherically symmetric then .]
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Paper 3, Section II, B
Part IB, 2018 commentWhat is the physical significance of the expectation value
of an observable in the normalised state ? Let and be two observables. By considering the norm of for real values of , show that
Deduce the generalised uncertainty relation
where the uncertainty in the state is defined by
A particle of mass moves in one dimension under the influence of the potential . By considering the commutator , show that every energy eigenvalue satisfies
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Paper 2, Section II, B
Part IB, 2018 commentFor an electron in a hydrogen atom, the stationary-state wavefunctions are of the form , where in suitable units obeys the radial equation
Explain briefly how the terms in this equation arise.
This radial equation has bound-state solutions of energy , where . Show that when , there is a solution of the form , and determine . Find the expectation value in this state.
Determine the total degeneracy of the energy level with energy .
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Paper 1, Section I, H
Part IB, 2018 commentform a random sample from a distribution whose probability density function is
where the value of the positive parameter is unknown. Determine the maximum likelihood estimator of the median of this distribution.
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Paper 2, Section I,
Part IB, 2018 commentDefine a simple hypothesis. Define the terms size and power for a test of one simple hypothesis against another. State the Neyman-Pearson lemma.
There is a single observation of a random variable which has a probability density function . Construct a best test of size for the null hypothesis
against the alternative hypothesis
Calculate the power of your test.
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Paper 1, Section II, H
Part IB, 2018 comment(a) Consider the general linear model where is a known matrix, is an unknown vector of parameters, and is an vector of independent random variables with unknown variances . Show that, provided the matrix is of rank , the least squares estimate of is
Let
What is the distribution of ? Write down, in terms of , an unbiased estimator of .
(b) Four points on the ground form the vertices of a plane quadrilateral with interior angles , so that . Aerial observations are made of these angles, where the observations are subject to independent errors distributed as random variables.
(i) Represent the preceding model as a general linear model with observations and unknown parameters .
(ii) Find the least squares estimates .
(iii) Determine an unbiased estimator of . What is its distribution?
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Paper 4, Section II, H
Part IB, 2018 commentThere is widespread agreement amongst the managers of the Reliable Motor Company that the number of faulty cars produced in a month has a binomial distribution
where is the total number of cars produced in a month. There is, however, some dispute about the parameter . The general manager has a prior distribution for which is uniform, while the more pessimistic production manager has a prior distribution with density , both on the interval .
In a particular month, faulty cars are produced. Show that if the general manager's loss function is , where is her estimate and the true value, then her best estimate of is
The production manager has responsibilities different from those of the general manager, and a different loss function given by . Find his best estimate of and show that it is greater than that of the general manager unless .
[You may use the fact that for non-negative integers ,
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Paper 3, Section II, H
Part IB, 2018 commentA treatment is suggested for a particular illness. The results of treating a number of patients chosen at random from those in a hospital suffering from the illness are shown in the following table, in which the entries are numbers of patients.
Describe the use of Pearson's statistic in testing whether the treatment affects recovery, and outline a justification derived from the generalised likelihood ratio statistic. Show that
[Hint: You may find it helpful to observe that
Comment on the use of this statistical technique when
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Paper 1, Section I, B
Part IB, 2018 commentFind, using a Lagrange multiplier, the four stationary points in of the function subject to the constraint . By sketching sections of the constraint surface in each of the coordinate planes, or otherwise, identify the nature of the constrained stationary points.
How would the location of the stationary points differ if, instead, the function were subject to the constraint
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Paper 3, Section I, B
Part IB, 2018 commentFor a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is
Determine the equations of motion. Show that is a conserved quantity, and use this result to simplify the equation of motion for . Deduce that
is a conserved quantity. What is the interpretation of ?
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Paper 2, Section II, B
Part IB, 2018 commentDerive the Euler-Lagrange equation for the integral
when and take given values at the fixed endpoints.
Show that the only function with and as for which the integral
is stationary is .
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Paper 4, Section II, B
Part IB, 2018 comment(a) A two-dimensional oscillator has action
Find the equations of motion as the Euler-Lagrange equations associated with , and use them to show that
is conserved. Write down the general solution of the equations of motion in terms of and , and evaluate in terms of the coefficients that arise in the general solution.
(b) Another kind of oscillator has action
where and are real constants. Find the equations of motion and use these to show that in general is not conserved. Find the special value of the ratio for which is conserved. Explain what is special about the action in this case, and state the interpretation of .
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Paper 4, Section II, I
Part II, 2018 commentState a theorem which describes the canonical divisor of a smooth plane curve in terms of the divisor of a hyperplane section. Express the degree of the canonical divisor and the genus of in terms of the degree of . [You need not prove these statements.]
From now on, we work over . Consider the curve in defined by the equation
Let be its projective completion. Show that is smooth.
Compute the genus of by applying the Riemann-Hurwitz theorem to the morphism induced from the rational map . [You may assume that the discriminant of is .]
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Paper 3, Section II, I
Part II, 2018 comment(a) State the Riemann-Roch theorem.
(b) Let be a smooth projective curve of genus 1 over an algebraically closed field , with char . Show that there exists an isomorphism from to the plane cubic in defined by the equation
for some distinct .
(c) Let be the point at infinity on . Show that the map is an isomorphism.
Describe how this defines a group structure on . Denote addition by . Determine all the points with in terms of the equation of the plane curve in part (b).
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Paper 2, Section II, I
Part II, 2018 comment(a) Let be an affine algebraic variety defined over the field .
Define the tangent space for , and the dimension of in terms of .
Suppose that is an algebraically closed field with char . Show directly from your definition that if , where is irreducible, then .
[Any form of the Nullstellensatz may be used if you state it clearly.]
(b) Suppose that char , and let be the vector space of homogeneous polynomials of degree in 3 variables over . Show that
is a non-empty Zariski open subset of .
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Paper 1, Section II, I
Part II, 2018 comment(a) Let be an uncountable field, a maximal ideal and
Show that every element of is algebraic over .
(b) Now assume that is algebraically closed. Suppose that is an ideal, and that vanishes on . Using the result of part (a) or otherwise, show that for some .
(c) Let be a morphism of affine algebraic varieties. Show if and only if the map is injective.
Suppose now that , and that and are irreducible. Define the dimension of , and show . [You may use whichever definition of you find most convenient.]
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Paper 3, Section II, H
Part II, 2018 comment(a) State a version of the Seifert-van Kampen theorem for a cell complex written as the union of two subcomplexes .
(b) Let
for , and take any . Write down a presentation for .
(c) By computing a homology group of a suitable four-sheeted covering space of , prove that is not homotopy equivalent to a compact, connected surface whenever .
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Paper 2, Section II, H
Part II, 2018 comment(a) Define the first barycentric subdivision of a simplicial complex . Hence define the barycentric subdivision . [You do not need to prove that is a simplicial complex.]
(b) Define the mesh of a simplicial complex . State a result that describes the behaviour of as .
(c) Define a simplicial approximation to a continuous map of polyhedra
Prove that, if is a simplicial approximation to , then the realisation is homotopic to .
(d) State and prove the simplicial approximation theorem. [You may use the Lebesgue number lemma without proof, as long as you state it clearly.]
(e) Prove that every continuous map of spheres is homotopic to a constant map when .
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Paper 1, Section II, H
Part II, 2018 comment(a) Let be the vector space of 3-dimensional upper-triangular matrices with real entries:
Let be the set of elements of for which are integers. Notice that is a subgroup of ; let act on by left-multiplication and let . Show that the quotient is a covering map.
(b) Consider the unit circle , and let . Show that the map defined by
is a homeomorphism.
(c) Let , where is the smallest equivalence relation satisfying
for all . Prove that and are homeomorphic by exhibiting a homeomorphism . [You may assume without proof that is Hausdorff.]
(d) Prove that .
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Paper 4, Section II, H
Part II, 2018 comment(a) State the Mayer-Vietoris theorem for a union of simplicial complexes
with .
(b) Construct the map that appears in the statement of the theorem. [You do not need to prove that the map is well defined, or a homomorphism.]
(c) Let be a simplicial complex with homeomorphic to the -dimensional sphere , for . Let be a subcomplex with homeomorphic to . Suppose that , such that has polyhedron identified with . Prove that has two path components.
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Paper 3, Section II,
Part II, 2018 comment(a) Let be a measure space. Define the spaces for . Prove that if then for all .
(b) Now let endowed with Borel sets and Lebesgue measure. Describe the dual spaces of for . Define reflexivity and say which are reflexive. Prove that is not the dual space of
(c) Now let be a Borel subset and consider the measure space induced from Borel sets and Lebesgue measure on .
(i) Given any , prove that any sequence in converging in to some admits a subsequence converging almost everywhere to .
(ii) Prove that if for then . [Hint: You might want to prove first that the inclusion is continuous with the help of one of the corollaries of Baire's category theorem.]
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Paper 1, Section II, F
Part II, 2018 comment(a) Consider a measure space and a complex-valued measurable function on . Prove that for any differentiable and increasing such that , then
where is the Lebesgue measure.
(b) Consider a complex-valued measurable function and its maximal function . Prove that for there is a constant such that .
[Hint: Split with and and prove that . Then use the maximal inequality for some constant
(c) Consider with and such that . Define and prove .
[Hint: Split the integral into and for all , given some suitable
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Paper 4, Section II, 23F
Part II, 2018 commentHere and below, is smooth such that and
denotes the set of continuously differentiable complex-valued functions with compact support on .
(a) Prove that there are constants and so that for any and :
[Hint: Denote , expand the square and integrate by parts.]
(b) Prove that, given any , there is a so that for any with :
[Hint: Use the fundamental theorem of calculus to control the second term of the left-hand side, and then compare to its weighted mean to control the first term of the left-hand side.]
(c) Prove that, given any , there is a so that for any :
[Hint: Show first that one can reduce to the case . Then argue by contradiction with the help of the Arzelà-Ascoli theorem and part (b).]
(d) Deduce that there is a so that for any :
[Hint: Show first that one can reduce to the case . Then combine the inequality (a), multiplied by a constant of the form (where is chosen so that be sufficiently small), and the inequality (c).]
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Paper 1, Section II, A
Part II, 2018 commentA particle of mass moves in one dimension in a periodic potential satisfying . Define the Floquet matrix . Show that and explain why Tr is real. Show that allowed bands occur for energies such that . Consider the potential
For states of negative energy, construct the Floquet matrix with respect to the basis of states . Derive an inequality for the values of in an allowed energy band.
For states of positive energy, construct the Floquet matrix with respect to the basis of states . Derive an inequality for the values of in an allowed energy band.
Show that the state with zero energy lies in a forbidden region for .
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Paper 4, Section II, A
Part II, 2018 commentDefine a Bravais lattice in three dimensions. Define the reciprocal lattice . Define the Brillouin zone.
An FCC lattice has a basis of primitive vectors given by
where is an orthonormal basis of . Find a basis of reciprocal lattice vectors. What is the volume of the Brillouin zone?
The asymptotic wavefunction for a particle, of wavevector , scattering off a potential is
where and is the scattering amplitude. Give a formula for the Born approximation to the scattering amplitude.
Scattering of a particle off a single atom is modelled by a potential with -function support on a spherical shell, centred at the origin. Calculate the Born approximation to the scattering amplitude, denoting the resulting expression as .
Scattering of a particle off a crystal consisting of atoms located at the vertices of a lattice is modelled by a potential
where as above. Calculate the Born approximation to the scattering amplitude giving your answer in terms of your approximate expression for scattering off a single atom. Show that the resulting amplitude vanishes unless the momentum transfer lies in the reciprocal lattice .
For the particular FCC lattice considered above, show that, when , scattering occurs for two values of the scattering angle, and , related by
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Paper 3, Section II, A
Part II, 2018 commentA beam of particles of mass and momentum is incident along the -axis. The beam scatters off a spherically symmetric potential . Write down the asymptotic form of the wavefunction in terms of the scattering amplitude .
The incoming plane wave and the scattering amplitude can be expanded in partial waves as,
where are Legendre polynomials. Define the -matrix. Assuming that the S-matrix is unitary, explain why we can write
for some real phase shifts . Obtain an expression for the total cross-section in terms of the phase shifts .
[Hint: You may use the orthogonality of Legendre polynomials:
Consider the repulsive, spherical potential
where . By considering the s-wave solution to the Schrödinger equation, show that
For low momenta, , compute the s-wave contribution to the total cross-section. Comment on the physical interpretation of your result in the limit .
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Paper 2, Section II, A
Part II, 2018 commentConsider a one-dimensional chain of atoms, each of mass . Impose periodic boundary conditions. The forces between neighbouring atoms are modelled as springs, with alternating spring constants and . In equilibrium, the separation between the atoms is .
Denote the position of the atom as . Let be the displacement from equilibrium. Write down the equations of motion of the system.
Show that the longitudinal modes of vibration are labelled by a wavenumber that is restricted to lie in a Brillouin zone. Find the frequency spectrum. What is the frequency gap at the edge of the Brillouin zone? Show that the gap vanishes when . Determine approximations for the frequencies near the centre of the Brillouin zone. Plot the frequency spectrum. What is the speed of sound in this system?
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Paper 4, Section II, J
Part II, 2018 commentLet be independent, identically distributed random variables with finite mean . Explain what is meant by saying that the random variable is a stopping time with respect to the sequence .
Let be a stopping time with finite mean . Prove Wald's equation:
[Here and in the following, you may use any standard theorem about integration.]
Suppose the are strictly positive, and let be the renewal process with interarrival times . Prove that satisfies the elementary renewal theorem:
A computer keyboard contains 100 different keys, including the lower and upper case letters, the usual symbols, and the space bar. A monkey taps the keys uniformly at random. Find the mean number of keys tapped until the first appearance of the sequence 'lava' as a sequence of 4 consecutive characters.
Find the mean number of keys tapped until the first appearance of the sequence 'aa' as a sequence of 2 consecutive characters.
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Paper 3, Section II, J
Part II, 2018 commentIndividuals arrive in a shop in the manner of a Poisson process with intensity , and they await service in the order of their arrival. Their service times are independent, identically distributed random variables . For , let be the number remaining in the shop immediately after the th departure. Show that
where is the number of arrivals during the th service period, and .
Show that
where is a typical service period, and is the traffic intensity of the queue.
Suppose , and the queue is in equilibrium in the sense that and have the same distribution for all . Express in terms of . Deduce that the mean waiting time (prior to service) of a typical individual is .
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Paper 2, Section II, J
Part II, 2018 commentLet be a continuous-time Markov chain on the finite state space . Define the terms generator (or Q-matrix) and invariant distribution, and derive an equation that links the generator and any invariant distribution . Comment on the possible non-uniqueness of invariant distributions.
Suppose is irreducible, and let be a Poisson process with intensity , that is independent of . Let be the value of immediately after the th arrival-time of (and . Show that is a discrete-time Markov chain, state its transition matrix and prove that it has the same invariant distribution as .
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Paper 1, Section II, J
Part II, 2018 commentLet be a continuous function. Explain what is meant by an inhomogeneous Poisson process with intensity function .
Let be such an inhomogeneous Poisson process, and let where is strictly increasing, differentiable and satisfies . Show that is a homogeneous Poisson process with intensity 1 if for all , where . Deduce that has the Poisson distribution with mean .
Bicycles arrive at the start of a long road in the manner of a Poisson process with constant intensity . The th bicycle has constant velocity , where are independent, identically distributed random variables, which are independent of . Cyclists can overtake one another freely. Show that the number of bicycles on the first miles of the road at time has the Poisson distribution with parameter .
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Paper 2, Section II, B
Part II, 2018 commentGiven that obtain the value of for real positive . Also obtain the value of , for real positive , in terms of
For , let
Find the leading terms in the asymptotic expansions as of (i) with fixed, and (ii) of .
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Paper 3, Section II, B
Part II, 2018 comment(a) Find the curves of steepest descent emanating from for the integral
for and determine the angles at which they meet at , and their asymptotes at infinity.
(b) An integral representation for the Bessel function for real is
Show that, as , with fixed,
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Paper 4, Section II, B
Part II, 2018 commentShow that
is a solution to the equation
and obtain the first two terms in the asymptotic expansion of as .
For , define a new dependent variable , and show that if solves the preceding equation then
Obtain the Liouville-Green approximate solutions to this equation for large positive , and compare with your asymptotic expansion for at the leading order.
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Paper 4, Section I, 4G
Part II, 2018 comment(a) State the theorem, the recursion theorem, and Rice's theorem.
(b) Show that if is partial recursive, then there is some such that
(c) By considering the partial function given by
show there exists some such that has exactly elements.
(d) Given , is it possible to compute whether or not has exactly 9 elements? Justify your answer.
[Note that we define . Any use of Church's thesis in your answers should be explicitly stated.]
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Paper 3, Section I, G
Part II, 2018 comment(a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form ( .
(b) Give an algorithm for converting a CFG into a corresponding CFG in CNF satisfying . [You need only outline the steps, without proof.]
(c) Convert the following :
into a grammar in CNF.
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Paper 2, Section I, G
Part II, 2018 comment(a) Let be a nondeterministic finite-state automaton with -transitions -NFA). Define the deterministic finite-state automaton (DFA) obtained from via the subset construction with -transitions.
(b) Let and be as above. By inducting on lengths of words, prove that
(c) Deduce that .
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Paper 1, Section I, G
Part II, 2018 comment(a) State the pumping lemma for context-free languages (CFLs).
(b) Which of the following are CFLs? Justify your answers.
(i)
(ii) and
(iii)
(c) Let be a CFL. Show that is also a CFL.
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Paper 3, Section II, G
Part II, 2018 comment(a) State and prove the pumping lemma for regular languages.
(b) Let be a minimal deterministic finite-state automaton whose language is finite. Let be the transition diagram of , and suppose there exists a non-empty closed path in starting and ending at state .
(i) Show that there is no path in from to any accept state of .
(ii) Show that there is no path in from to any other state of .
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Paper 1, Section II, G
Part II, 2018 comment(a) Define the halting set . Prove that is recursively enumerable, but not recursive.
(b) Given , define a many-one reduction of to . Show that if is recursively enumerable and , then is also recursively enumerable.
(c) Show that each of the functions and are both partial recursive and total, by building them up as partial recursive functions.
(d) Let . We define the set via
(i) Show that both and .
(ii) Using the above, or otherwise, give an explicit example of a subset of for which neither nor are recursively enumerable.
(iii) For every , show that if and then .
[Note that we define . Any use of Church's thesis in your answers should be explicitly stated.]
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Paper 1, Section I, B
Part II, 2018 commentDerive Hamilton's equations from an action principle.
Consider a two-dimensional phase space with the Hamiltonian . Show that is the first integral for some constant which should be determined. By considering the surfaces of constant in the extended phase space, solve Hamilton's equations, and sketch the orbits in the phase space.
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Paper 2, Section I, B
Part II, 2018 commentLet . Consider a Lagrangian
of a particle constrained to move on a sphere of radius . Use Lagrange multipliers to show that
Now, consider the system with , and find the particle trajectories.
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Paper 3, Section I, B
Part II, 2018 commentThree particles of unit mass move along a line in a potential
where is the coordinate of the th particle, .
Write the Lagrangian in the form
and specify the matrices and .
Find the normal frequencies and normal modes for this system.
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Paper 4, Section I, B
Part II, 2018 commentState and prove Noether's theorem in Lagrangian mechanics.
Consider a Lagrangian
for a particle moving in the upper half-plane in a potential which only depends on . Find two independent first integrals.
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Paper 2, Section II, B
Part II, 2018 commentDefine a body frame of a rotating rigid body, and show that there exists a vector such that
Let be the angular momentum of a free rigid body expressed in the body frame. Derive the Euler equations from the conservation of angular momentum.
Verify that the kinetic energy , and the total angular momentum are conserved. Hence show that
where is a quartic polynomial which should be explicitly determined in terms of and .
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Paper 4, Section II, B
Part II, 2018 commentGiven a Lagrangian with degrees of freedom , define the Hamiltonian and show how Hamilton's equations arise from the Lagrange equations and the Legendre transform.
Consider the Lagrangian for a symmetric top moving in constant gravity:
where and are constants. Construct the corresponding Hamiltonian, and find three independent Poisson-commuting first integrals of Hamilton's equations.
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Paper 4, Section I, H
Part II, 2018 commentWhat is a linear feedback shift register? Explain the Berlekamp-Massey method for recovering a feedback polynomial of a linear feedback shift register from its output. Illustrate the method in the case when we observe output
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Paper 3, Section , H
Part II, 2018 commentCompute the rank and minimum distance of the cyclic code with generator polynomial and parity check polynomial . Now let be a root of in the field with 8 elements. We receive the word . Verify that , and hence decode using minimum-distance decoding.
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Paper 2, Section , H
Part II, 2018 commentWhat is the channel matrix of a binary symmetric channel with error probability ?
State the maximum likelihood decoding rule and the minimum distance decoding rule. Prove that if , then they agree.
Let be the repetition code . Suppose a codeword from is sent through a binary symmetric channel with error probability . Show that, if the minimum distance decoding rule is used, then the probability of error is .
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Paper 1, Section , H
Part II, 2018 commentState and prove Shannon's noiseless coding theorem. [You may use Gibbs' and Kraft's inequalities as long as they are clearly stated.]
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Paper 1, Section II, H
Part II, 2018 commentDefine the bar product of binary linear codes and , where is a subcode of . Relate the rank and minimum distance of to those of and and justify your answer.
What is a parity check matrix for a linear code? If has parity check matrix and has parity check matrix , find a parity check matrix for .
Using the bar product construction, or otherwise, define the Reed-Muller code for . Compute the rank of . Show that all but two codewords in have the same weight. Given , for which is it true that all elements of have even weight? Justify your answer.
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Paper 2, Section II, H
Part II, 2018 commentDescribe the RSA encryption scheme with public key and private key .
Suppose with and distinct odd primes and with and coprime. Denote the order of in by . Further suppose divides where is odd. If prove that there exists such that the greatest common divisor of and is a nontrivial factor of . Further, prove that the number of satisfying is .
Hence, or otherwise, prove that finding the private key from the public key is essentially as difficult as factoring .
Suppose a message is sent using the scheme with and , and is the received text. What is ?
An integer satisfying is called a fixed point if it is encrypted to itself. Prove that if is a fixed point then so is .
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Paper 2, Section I, B
Part II, 2018 comment(a) Consider a homogeneous and isotropic universe with a uniform distribution of galaxies. For three galaxies at positions , show that spatial homogeneity implies that their non-relativistic velocities must satisfy
and hence that the velocity field coordinates are linearly related to the position coordinates via
where the matrix coefficients are independent of the position. Show why isotropy then implies Hubble's law
Explain how the velocity of a galaxy is determined by the scale factor and express the Hubble parameter today in terms of .
(b) Define the cosmological horizon . For an Einstein-de Sitter universe with , calculate at today in terms of . Briefly describe the horizon problem of the standard cosmology.
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Paper 3, Section I, B
Part II, 2018 commentThe energy density of a particle species is defined by
where is the energy, and the distribution function, of a particle with momentum . Here is the speed of light and is the rest mass of the particle. If the particle species is in thermal equilibrium then the distribution function takes the form
where is the number of degrees of freedom of the particle, is the temperature, and are constants and is for bosons and is for fermions.
(a) Stating any assumptions you require, show that in the very early universe the energy density of a given particle species is
(b) Show that the total energy density in the very early universe is
where is defined by
[Hint: You may use the fact that
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Paper 1, Section I, B
Part II, 2018 commentFor a homogeneous and isotropic universe filled with pressure-free matter , the Friedmann and Raychaudhuri equations are, respectively,
with mass density , curvature , and where . Using conformal time with , show that the relative density parameter can be expressed as
where and is the critical density of a flat universe (Einstein-de Sitter). Use conformal time again to show that the Friedmann and Raychaudhuri equations can be re-expressed as
From these derive the evolution equation for the density parameter :
Plot the qualitative behaviour of as a function of time relative to the expanding Einsteinde Sitter model with (i.e., include curves initially with and ).
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Paper 4, Section I, B
Part II, 2018 commentA constant overdensity is created by taking a spherical region of a flat matterdominated universe with radius and compressing it into a region with radius . The evolution is governed by the parametric equations
where is a constant and
where is the Hubble constant and is the fractional overdensity at time .
Show that, as ,
where the scale factor is given by .
that, when the spherical overdensity has collapsed to zero radius, the linear perturbation has value . -
Paper 3, Section II, B
Part II, 2018 commentThe pressure support equation for stars is
where is the density, is the pressure, is the radial distance, and is Newton's constant.
(a) What two boundary conditions should we impose on the above equation for it to describe a star?
(b) By assuming a polytropic equation of state,
where is a constant, derive the Lane-Emden equation
where , with the density at the centre of the star, and , for some that you should determine.
(c) Show that the mass of a polytropic star is
where and is the value of at the surface of the star.
(d) Derive the following relation between the mass, , and radius, , of a polytropic star
where you should determine the constant . What type of star does the polytrope represent and what is the significance of the mass being constant for this star?
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Paper 1, Section II, B
Part II, 2018 commentA flat homogeneous and isotropic universe with scale factor is filled with a scalar field with potential . Its evolution satisfies the Friedmann and scalar field equations,
where is the Hubble parameter, is the reduced Planck mass, and dots denote derivatives with respect to cosmic time , e.g. .
(a) Use these equations to derive the Raychaudhuri equation, expressed in the form:
(b) Consider the following ansatz for the scalar field evolution,
where are constants. Find the specific cosmological solution,
(c) Hence, show that the Hubble parameter can be expressed in terms of as
and that the scalar field ansatz solution ( ) requires the following form for the potential:
(d) Assume that the given parameters in are such that . Show that the asymptotic limit for the cosmological solution as exhibits decelerating power law evolution and that there is an accelerating solution as , that is,
Find the time at which the solution transitions from deceleration to acceleration.
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Paper 4, Section II, I
Part II, 2018 commentLet be a surface.
(a) Define what it means for a curve to be a geodesic, where and .
(b) A geodesic is said to be maximal if any geodesic with and satisfies . A surface is said to be geodesically complete if all maximal geodesics are defined on , otherwise, the surface is said to be geodesically incomplete. Give an example, with justification, of a non-compact geodesically complete surface which is not a plane.
(c) Assume that along any maximal geodesic
the following holds:
Here denotes the Gaussian curvature of .
(i) Show that is inextendible, i.e. if is a connected surface with , then .
(ii) Give an example of a surface which is geodesically incomplete and satisfies . Do all geodesically incomplete inextendible surfaces satisfy ? Justify your answer.
[You may use facts about geodesics from the course provided they are clearly stated.]
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Paper 3, Section II, I
Part II, 2018 commentLet be a surface.
(a) Define the Gaussian curvature of in terms of the coefficients of the first and second fundamental forms, computed with respect to a local parametrization of .
Prove the Theorema Egregium, i.e. show that the Gaussian curvature can be expressed entirely in terms of the coefficients of the first fundamental form and their first and second derivatives with respect to and .
(b) State the global Gauss-Bonnet theorem for a compact orientable surface .
(c) Now assume that is non-compact and diffeomorphic to but that there is a point such that is a compact subset of . Is it necessarily the case that Justify your answer.
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Paper 2, Section II, I
Part II, 2018 commentLet denote a regular curve.
(a) Show that there exists a parametrization of by arc length.
(b) Under the assumption that the curvature is non-zero, define the torsion of . Give an example of two curves and in whose curvature (as a function of arc length ) coincides and is non-vanishing, but for which the curves are not related by a rigid motion, i.e. such that is not identically where and
(c) Give an example of a simple closed curve , other than a circle, which is preserved by a non-trivial rigid motion, i.e. which satisfies
for some choice of with . Justify your answer.
(d) Now show that a simple closed curve which is preserved by a nontrivial smooth 1-parameter family of rigid motions is necessarily a circle, i.e. show the following:
Let be a regular curve. If for all ,
then is a circle. [You may use the fact that the set of fixed points of a non-trivial rigid motion is either or a line .]
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Paper 1, Section II, I
Part II, 2018 comment(a) Let be a manifold and . Define the tangent space and show that it is a vector subspace of , independent of local parametrization, of dimension equal to .
(b) Now show that depends continuously on in the following sense: if is a sequence in such that , and is a sequence such that , then . If , show that all arise as such limits where is a sequence in .
(c) Consider the set defined by , where . Show that, for all , the set is a smooth manifold. Compute its dimension.
(d) For as above, does depend continuously on and for all ? In other words, let be sequences with . Suppose that and . Is it necessarily the case that ? Justify your answer.
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Paper 1, Section II, E
Part II, 2018 commentConsider the system
where is a constant.
(a) Find and classify the fixed points of the system. For show that the linear classification of the non-hyperbolic fixed points is nonlinearly correct. For show that there are no periodic orbits. [Standard results for periodic orbits may be quoted without proof.]
(b) Sketch the phase plane for the cases (i) , (ii) , and (iii) , showing any separatrices clearly.
(c) For what values of a do stationary bifurcations occur? Consider the bifurcation with . Let be the values of at which the bifurcation occurs, and define . Assuming that , find the extended centre manifold to leading order. Further, determine the evolution equation on the centre manifold to leading order. Hence identify the type of bifurcation.
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Paper 4, Section II, E
Part II, 2018 commentLet be a continuous one-dimensional map of an interval . Define what it means (i) for to have a horseshoe (ii) for to be chaotic. [Glendinning's definition should be used throughout this question.]
Prove that if has a 3 -cycle then is chaotic. [You may assume the intermediate value theorem and any corollaries of it.]
State Sharkovsky's theorem.
Use the above results to deduce that if has an -cycle, where is any integer that is not a power of 2 , then is chaotic.
Explain briefly why if is chaotic then has -cycles for many values of that are not powers of 2. [You may assume that a map with a horseshoe acts on some set like the Bernoulli shift map acts on .]
The logistic map is not chaotic when and it has 3 -cycles when . What can be deduced from these statements about the values of for which the logistic map has a 10-cycle?
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Paper 3, Section II, 32E
Part II, 2018 commentConsider the system
where and are parameters.
By considering a function of the form , show that when the origin is globally asymptotically stable. Sketch the phase plane for this case.
Find the fixed points for the general case. Find the values of and for which the fixed points have (i) a stationary bifurcation and (ii) oscillatory (Hopf) bifurcations. Sketch these bifurcation values in the -plane.
For the case , find the leading-order approximation to the extended centre manifold of the bifurcation as varies, assuming that . Find also the evolution equation on the extended centre manifold to leading order. Deduce the type of bifurcation, and sketch the bifurcation diagram in the -plane.
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Paper 2, Section II, 32E
Part II, 2018 commentConsider the system
where and are real constants, and . Find and classify the fixed points.
Show that when the system is Hamiltonian and find . Sketch the phase plane for this case.
Suppose now that . Show that the small change in following a trajectory of the perturbed system around an orbit of the unperturbed system is given to leading order by an equation of the form
where should be found explicitly, and where and are the minimum and maximum values of on the unperturbed orbit.
Use the energy-balance method to find the value of , correct to leading order in , for which the system has a homoclinic orbit. [Hint: The substitution may prove useful.]
Over what range of would you expect there to be periodic solutions that enclose only one of the fixed points?
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Paper 1, Section II, D
Part II, 2018 commentDefine the field strength tensor for an electromagnetic field specified by a 4-vector potential . How do the components of change under a Lorentz transformation? Write down two independent Lorentz-invariant quantities which are quadratic in the field strength tensor.
[Hint: The alternating tensor takes the values and when is an even or odd permutation of respectively and vanishes otherwise. You may assume this is an invariant tensor of the Lorentz group. In other words, its components are the same in all inertial frames.]
In an inertial frame with spacetime coordinates , the 4-vector potential has components and the electric and magnetic fields are given as
Evaluate the components of in terms of the components of and . Show that the quantities
are the same in all inertial frames.
A relativistic particle of mass , charge and 4 -velocity moves according to the Lorentz force law,
Here is the proper time. For the case of a constant, uniform field, write down a solution of giving in terms of its initial value as an infinite series in powers of the field strength.
Suppose further that the fields are such that both and defined above are zero. Work in an inertial frame with coordinates where the particle is at rest at the origin at and the magnetic field points in the positive -direction with magnitude . The electric field obeys . Show that the particle moves on the curve for some constant which you should determine.
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Paper 4 , Section II, D
Part II, 2018 comment(a) Define the polarisation of a dielectric material and explain what is meant by the term bound charge.
Consider a sample of material with spatially dependent polarisation occupying a region with surface . Show that, in the absence of free charge, the resulting scalar potential can be ascribed to bulk and surface densities of bound charge.
Consider a sphere of radius consisting of a dielectric material with permittivity surrounded by a region of vacuum. A point-like electric charge is placed at the centre of the sphere. Determine the density of bound charge on the surface of the sphere.
(b) Define the magnetization of a material and explain what is meant by the term bound current.
Consider a sample of material with spatially-dependent magnetization occupying a region with surface . Show that, in the absence of free currents, the resulting vector potential can be ascribed to bulk and surface densities of bound current.
Consider an infinite cylinder of radius consisting of a material with permeability surrounded by a region of vacuum. A thin wire carrying current is placed along the axis of the cylinder. Determine the direction and magnitude of the resulting bound current density on the surface of the cylinder. What is the magnetization on the surface of the cylinder?
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Paper 3, Section II, D
Part II, 2018 commentStarting from the covariant form of the Maxwell equations and making a suitable choice of gauge which you should specify, show that the 4-vector potential due to an arbitrary 4-current obeys the wave equation,
where .
Use the method of Green's functions to show that, for a localised current distribution, this equation is solved by
for some that you should specify.
A point particle, of charge , moving along a worldline parameterised by proper time , produces a 4 -vector potential
where . Define and draw a spacetime diagram to illustrate its physical significance.
Suppose the particle follows a circular trajectory,
(with ), in some inertial frame with coordinates . Evaluate the resulting 4 -vector potential at a point on the -axis as a function of and .
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Paper 2, Section II, C
Part II, 2018 commentAn initially unperturbed two-dimensional inviscid jet in has uniform speed in the direction, while the surrounding fluid is stationary. The unperturbed velocity field is therefore given by
Consider separately disturbances in which the layer occupies varicose disturbances) and disturbances in which the layer occupies sinuous disturbances , where , and determine the dispersion relation in each case.
Find asymptotic expressions for the real part of in the limits and and draw sketches of in each case.
Compare the rates of growth of the two types of disturbance.
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Paper 1, Section II, C
Part II, 2018 commentA two-dimensional layer of very viscous fluid of uniform thickness sits on a stationary, rigid surface . It is impacted by a stream of air (which can be assumed inviscid) such that the air pressure at is , where and are constants, is the density of the air, and is the coordinate parallel to the surface.
What boundary conditions apply to the velocity and stress tensor of the viscous fluid at and ?
By assuming the form for the stream function of the flow, or otherwise, solve the Stokes equations for the velocity and pressure fields. Show that the layer thins at a rate
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Paper 4, Section II, C
Part II, 2018 commentA cylinder of radius rotates about its axis with angular velocity while its axis is fixed parallel to and at a distance from a rigid plane, where . Fluid of kinematic viscosity fills the space between the cylinder and the plane. Determine the gap width between the cylinder and the plane as a function of a coordinate parallel to the surface of the wall and orthogonal to the axis of the cylinder. What is the characteristic length scale, in the direction, for changes in the gap width? Taking an appropriate approximation for , valid in the region where the gap width is small, use lubrication theory to determine that the volume flux between the wall and the cylinder (per unit length along the axis) has magnitude , and state its direction.
Evaluate the tangential shear stress on the surface of the cylinder. Approximating the torque on the cylinder (per unit length along the axis) in the form of an integral , find the torque to leading order in .
Explain the restriction for the theory to be valid.
[You may use the facts that and
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Paper 3, Section II, C
Part II, 2018 commentFor two Stokes flows and inside the same volume with different boundary conditions on its boundary , prove the reciprocal theorem
where and are the stress tensors associated with the flows.
Stating clearly any properties of Stokes flow that you require, use the reciprocal theorem to prove that the drag on a body translating with uniform velocity is given by
where is a symmetric second-rank tensor that depends only on the geometry of the body.
A slender rod falls slowly through very viscous fluid with its axis inclined to the vertical. Explain why the rod does not rotate, stating any properties of Stokes flow that you use.
When the axis of the rod is inclined at an angle to the vertical, the centre of mass of the rod travels at an angle to the vertical. Given that the rod falls twice as quickly when its axis is vertical as when its axis is horizontal, show that
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Paper 1, Section I, B
Part II, 2018 commentThe Beta and Gamma functions are defined by
where .
(a) By using a suitable substitution, or otherwise, prove that
for . Extending by analytic continuation, for which values of does this result hold?
(b) Prove that
for
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Paper 2, Section ,
Part II, 2018 commentShow that
in the sense of Cauchy principal value, where and are positive integers. [State clearly any standard results involving contour integrals that you use.]
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Paper 3, Section I, B
Part II, 2018 commentUsing a suitable branch cut, show that
where .
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Paper 4, Section I, B
Part II, 2018 commentState the conditions for a point to be a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane.
Find all singular points of the Bessel equation
and determine whether they are regular or irregular.
By writing , find two linearly independent solutions of . Comment on the relationship of your solutions to the nature of the singular points.
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Paper 2, Section II, B
Part II, 2018 commentConsider a multi-valued function .
(a) Explain what is meant by a branch point and a branch cut.
(b) Consider .
(i) By writing , where , and , deduce the expression for in terms of and . Hence, show that is infinitely valued and state its principal value.
(ii) Show that and are the branch points of . Deduce that the line is a possible choice of branch cut.
(iii) Use the Cauchy-Riemann conditions to show that is analytic in the cut plane. Show that .
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Paper 1, Section II, B
Part II, 2018 commentThe equation
where is a constant with , has solutions of the form
for suitably chosen contours and some suitable function .
(a) Find and determine the condition on , which you should express in terms of and .
(b) Use the results of part (a) to show that can be a finite contour and specify two possible finite contours with the help of a clearly labelled diagram. Hence, find the corresponding solution of the equation in the case .
(c) In the case and real , show that can be an infinite contour and specify two possible infinite contours with the help of a clearly labelled diagram. [Hint: Consider separately the cases and .] Hence, find a second, linearly independent solution of the equation ( ) in this case.
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Paper 4, Section II, I
Part II, 2018 commentLet be a field of characteristic and let be the splitting field of the polynomial over , where . Let be a root of .
If , show that is irreducible over , that , and that is a Galois extension of . What is ?
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Paper 3, Section II, I
Part II, 2018 commentLet be a finite field extension of a field , and let be a finite group of automorphisms of . Denote by the field of elements of fixed by the action of .
(a) Prove that the degree of over is equal to the order of the group .
(b) For any write .
(i) Suppose that . Prove that the coefficients of generate over .
(ii) Suppose that . Prove that the coefficients of and lie in . By considering the case with and in , or otherwise, show that they need not generate over .
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Paper 2, Section II, I
Part II, 2018 commentLet be a field and let be a monic polynomial with coefficients in . What is meant by a splitting field for over ? Show that such a splitting field exists and is unique up to isomorphism.
Now suppose that is a finite field. Prove that is a Galois extension of with cyclic Galois group. Prove also that the degree of over is equal to the least common multiple of the degrees of the irreducible factors of over .
Now suppose is the field with two elements, and let
How many elements does the set have?
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Paper 1, Section II, I
Part II, 2018 commentLet be an irreducible quartic with rational coefficients. Explain briefly why it is that if the cubic has as its Galois group then the Galois group of is .
For which prime numbers is the Galois group of a proper subgroup of ? [You may assume that the discriminant of is .]
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Paper 1, Section II, 37E
Part II, 2018 commentConsider the de Sitter metric
where is a constant.
(a) Write down the Lagrangian governing the geodesics of this metric. Use the Euler-Lagrange equations to determine all non-vanishing Christoffel symbols.
(b) Let be a timelike geodesic parametrized by proper time with initial conditions at ,
where the dot denotes differentiation with respect to and is a constant. Assuming both and to be future oriented, show that at ,
(c) Find a relation between and along the geodesic of part (b) and show that for a finite value of . [You may use without proof that
(d) Briefly interpret this result.
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Paper 2, Section II, E
Part II, 2018 commentThe Friedmann equations and the conservation of energy-momentum for a spatially homogeneous and isotropic universe are given by:
where is the scale factor, the energy density, the pressure, the cosmological constant and .
(a) Show that for an equation of state constant, the energy density obeys , for some constant .
(b) Consider the case of a matter dominated universe, , with . Write the equation of motion for the scale factor in the form of an effective potential equation,
where you should determine the constant and the potential . Sketch the potential together with the possible values of and qualitatively discuss the long-term dynamics of an initially small and expanding universe for the cases .
(c) Repeat the analysis of part (b), again assuming , for the cases:
(i) ,
(ii) ,
(iii) .
Discuss all qualitatively different possibilities for the dynamics of the universe in each case.
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Paper 4, Section II, E
Part II, 2018 comment(a) In the Newtonian weak-field limit, we can write the spacetime metric in the form
where and the potential , as well as the velocity of particles moving in the gravitational field are assumed to be small, i.e.,
Use the geodesic equation for this metric to derive the equation of motion for a massive point particle in the Newtonian limit.
(b) The far-field limit of the Schwarzschild metric is a special case of (*) given, in spherical coordinates, by
where now . For the following questions, state your results to first order in , i.e. neglecting terms of .
(i) Let . Calculate the proper length along the radial curve from to at fixed .
(ii) Consider a massless particle moving radially from to . According to an observer at rest at , what time elapses during this motion?
(iii) The effective velocity of the particle as seen by the observer at is defined as . Evaluate and then take the limit of this result as . Briefly discuss the value of in this limit.
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Paper 3, Section II, E
Part II, 2018 commentThe Schwarzschild metric in isotropic coordinates , is given by:
where
and is the mass of the black hole.
(a) Let , denote a coordinate system related to by
where and . Write down the transformation matrix , briefly explain its physical meaning and show that the inverse transformation is of the same form, but with .
(b) Using the coordinate transformation matrix of part (a), or otherwise, show that the components of the metric in coordinates are given by
where and are functions of that you should determine. You should also express in terms of the coordinates .
(c) Consider the limit with held constant. Show that for points the function , while tends to a finite value, which you should determine. Hence determine the metric components at points in this limit.
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Paper 4, Section II, I
Part II, 2018 commentLet . Define the Ramsey number . Show that exists and that .
Show that . Show that (up to relabelling the vertices) there is a unique way to colour the edges of the complete graph blue and yellow with no monochromatic triangle.
What is the least positive integer such that the edges of the complete graph can be coloured blue and yellow in such a way that there are precisely monochromatic triangles?
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Paper 3, Section II, I
Part II, 2018 commentWhat does it mean to say that a graph has a -colouring? What are the chromatic number and the independence number of a graph ? For each , give an example of a graph such that but .
Let . Show that there exists a graph containing no cycle of length with .
Show also that if is sufficiently large then there is a triangle-free of order with .
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Paper 2, Section II, I
Part II, 2018 commentLet be a graph and . Show that if every -separator in has order at least then there exist vertex-disjoint -paths in .
Let and assume that is -connected. Show that must contain a cycle of length at least .
Assume further that . Must contain a cycle of length at least Justify your answer.
What is the largest integer such that any 3-connected graph with must contain a cycle of length at least ?
[No form of Menger's theorem or of the max-flow-min-cut theorem may be assumed without proof.]
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Paper 1, Section II, I
Part II, 2018 comment(a) Define ex where is a graph with at least one edge and . Show that, for any such , the exists.
[You may not assume the Erdős-Stone theorem.]
(b) State the Erdős-Stone theorem. Use it to deduce that if is bipartite then
(c) Let . Show that ex .
We say is nice if whenever with then either or . Let is nice . Show that
denotes the set of integers modulo , i.e. with addition modulo
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Paper 1, Section II, A
Part II, 2018 commentLet be equipped with the standard symplectic form so that the Poisson bracket is given by:
for real-valued functions on . Let be a Hamiltonian function.
(a) Write down Hamilton's equations for , define a first integral of the system and state what it means that the system is integrable.
(b) State the Arnol'd-Liouville theorem.
(c) Define complex coordinates by , and show that if are realvalued functions on then:
(d) For an anti-Hermitian matrix with components , let . Show that:
where is the usual matrix commutator.
(e) Consider the Hamiltonian:
Show that is integrable and describe the invariant tori.
[In this question , and the summation convention is understood for these indices.]
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Paper 2, Section II, A
Part II, 2018 comment(a) Let be two families of linear operators, depending on a parameter , which act on a Hilbert space with inner product , . Suppose further that for each is self-adjoint and that is anti-self-adjoint. State 's equation for the pair , and show that if it holds then the eigenvalues of are independent of .
(b) For , define the inner product:
Let be the operators:
where are smooth, real-valued functions. You may assume that the normalised eigenfunctions of are smooth functions of , which decay rapidly as for all .
(i) Show that if are smooth and rapidly decaying towards infinity then:
Deduce that the eigenvalues of are real.
(ii) Show that if Lax's equation holds for , then must satisfy the Boussinesq equation:
where are constants whose values you should determine. [You may assume without proof that the identity:
holds for smooth, rapidly decaying
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Paper 3, Section II, A
Part II, 2018 commentSuppose is a smooth one-parameter group of transformations acting on .
(a) Define the generator of the transformation,
where you should specify and in terms of .
(b) Define the prolongation of and explicitly compute in terms of .
Recall that if is a Lie point symmetry of the ordinary differential equation:
then it follows that whenever .
(c) Consider the ordinary differential equation:
for a smooth function. Show that if generates a Lie point symmetry of this equation, then:
(d) Find all the Lie point symmetries of the equation:
where is an arbitrary smooth function.
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Paper 3, Section II, F
Part II, 2018 comment(a) Let be a normed vector space and let be a Banach space. Show that the space of bounded linear operators is a Banach space.
(b) Let and be Banach spaces, and let be a dense linear subspace. Prove that a bounded linear map can be extended uniquely to a bounded linear map with the same operator norm. Is the claim also true if one of and is not complete?
(c) Let be a normed vector space. Let be a sequence in such that
Prove that there is a constant such that
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Paper 1, Section II, F
Part II, 2018 commentLet be a compact Hausdorff space.
(a) State the Arzelà-Ascoli theorem, and state both the real and complex versions of the Stone-Weierstraß theorem. Give an example of a compact space and a bounded set of functions in that is not relatively compact.
(b) Let be continuous. Show that there exists a sequence of polynomials in variables such that
Characterize the set of continuous functions for which there exists a sequence of polynomials such that uniformly on .
(c) Prove that if is equicontinuous then is finite. Does this implication remain true if we drop the requirement that be compact? Justify your answer.
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Paper 2, Section II, F
Part II, 2018 commentLet be Banach spaces and let denote the space of bounded linear operators .
(a) Define what it means for a bounded linear operator to be compact. Let be linear operators with finite rank, i.e., is finite-dimensional. Assume that the sequence converges to in . Show that is compact.
(b) Let be compact. Show that the dual map is compact. [Hint: You may use the Arzelà-Ascoli theorem.]
(c) Let be a Hilbert space and let be a compact operator. Let be an infinite sequence of eigenvalues of with eigenvectors . Assume that the eigenvectors are orthogonal to each other. Show that .
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Paper 4, Section II, F
Part II, 2018 comment(a) Let be a separable normed space. For any sequence with for all , show that there is and a subsequence such that for all as . [You may use without proof the fact that is complete and that any bounded linear map , where is a dense linear subspace, can be extended uniquely to an element .]
(b) Let be a Hilbert space and a unitary map. Let
Prove that and are orthogonal, , and that for every ,
where is the orthogonal projection onto the closed subspace .
(c) Let be a linear map, where is the unit circle, induced by a homeomorphism by . Prove that there exists with such that for all . (Here denotes the function on which returns 1 identically.) If is not the identity map, does it follow that as above is necessarily unique? Justify your answer.
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Paper 4, Section II, G
Part II, 2018 commentState and prove the -Recursion Theorem. [You may assume the Principle of Induction.]
What does it mean to say that a relation on a set is well-founded and extensional? State and prove Mostowski's Collapsing Theorem. [You may use any recursion theorem from the course, provided you state it precisely.]
For which sets is it the case that every well-founded extensional relation on is isomorphic to the relation on some transitive subset of ?
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Paper 3, Section II, G
Part II, 2018 commentState and prove the Compactness Theorem for first-order predicate logic. State and prove the Upward Löwenheim-Skolem Theorem.
[You may assume the Completeness Theorem for first-order predicate logic.]
For each of the following theories, either give axioms (in the specified language) for the theory or prove that the theory is not axiomatisable.
(i) The theory of finite groups (in the language of groups).
(ii) The theory of groups in which every non-identity element has infinite order (in the language of groups).
(iii) The theory of total orders (in the language of posets).
(iv) The theory of well-orderings (in the language of posets).
If a theory is axiomatisable by a set of sentences, and also by a finite set of sentences, does it follow that the theory is axiomatisable by some finite subset of ? Justify your answer.
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Paper 2, Section II, G
Part II, 2018 commentState and prove the Knaster-Tarski Fixed-Point Theorem. Deduce the SchröderBernstein Theorem.
Show that the poset of all countable subsets of (ordered by inclusion) is not complete.
Find an order-preserving function that does not have a fixed point. [Hint: Start by well-ordering the reals.]
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Paper 1, Section II, G
Part II, 2018 commentGive the inductive definition of ordinal exponentiation. Use it to show that whenever (for ), and also that whenever for .
Give an example of ordinals and with such that .
Show that , for any ordinals , and give an example to show that we need not have .
For which ordinals do we have ? And for which do we have ? Justify your answers.
[You may assume any standard results not concerning ordinal exponentiation.]
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Paper 1, Section I,
Part II, 2018 commentConsider a birth-death process in which the birth and death rates in a population of size are, respectively, and , where and are per capita birth and death rates.
(a) Write down the master equation for the probability, , of the population having size at time .
(b) Obtain the differential equations for the rates of change of the mean and the variance in terms of and .
(c) Compare the equations obtained above with the deterministic description of the evolution of the population size, . Comment on why and cannot be uniquely deduced from the deterministic model but can be deduced from the stochastic description.
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Paper 2, Section I, C
Part II, 2018 commentConsider a model of an epidemic consisting of populations of susceptible, , infected, , and recovered, , individuals that obey the following differential equations
where and are constant. Show that the sum of susceptible, infected and recovered individuals is a constant . Find the fixed points of the dynamics and deduce the condition for an endemic state with a positive number of infected individuals. Expressing in terms of and , reduce the system of equations to two coupled differential equations and, hence, deduce the conditions for the fixed point to be a node or a focus. How do small perturbations of the populations relax to the steady state in each case?
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Paper 3, Section I,
Part II, 2018 commentConsider a nonlinear model for the axisymmetric dispersal of a population in two spatial dimensions whose density, , obeys
where is a positive constant, is a radial polar coordinate, and is time.
Show that
is constant. Interpret this condition.
Show that a similarity solution of the form
is valid for provided that the scaling function satisfies
Show that there exists a value (which need not be evaluated) such that for but for . Determine the area within which at time in terms of .
[Hint: The gradient and divergence operators in cylindrical polar coordinates act on radial functions and as
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Paper 4, Section I, C
Part II, 2018 commentConsider a model of a population in discrete time
where are constants and Interpret the constants and show that for there is a stable fixed point.
Suppose the initial condition is and that . Show, using a cobweb diagram, that the population is bounded as
and attains the bounds.
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Paper 3, Section II, C
Part II, 2018 commentConsider fluctuations of a population described by the vector . The probability of the state at time , obeys the multivariate Fokker-Planck equation
where is a drift vector and is a symmetric positive-definite diffusion matrix, and the summation convention is used throughout.
(a) Show that the Fokker-Planck equation can be expressed as a continuity equation
for some choice of probability flux which you should determine explicitly. Here, denotes the gradient operator.
(b) Show that the above implies that an initially normalised probability distribution remains normalised,
at all times, where the volume element .
(c) Show that the first two moments of the probability distribution obey
(d) Now consider small fluctuations with zero mean, and assume that it is possible to linearise the drift vector and the diffusion matrix as and where has real negative eigenvalues and is a symmetric positive-definite matrix. Express the probability flux in terms of the matrices and and assume that it vanishes in the stationary state.
(e) Hence show that the multivariate normal distribution,
where is a normalisation and is symmetric, is a solution of the linearised FokkerPlanck equation in the stationary state, and obtain an equation that relates to the matrices and .
(f) Show that the inverse of the matrix is the matrix of covariances and obtain an equation relating to the matrices and .
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Paper 4, Section II, C
Part II, 2018 commentAn activator-inhibitor reaction diffusion system is given, in dimensionless form, by
where and are positive constants. Which symbol represents the concentration of activator and which the inhibitor? Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the spatially uniform situation, that the reaction kinetics are stable if .
Determine the conditions for the steady state to be driven unstable by diffusion, and sketch the parameter space in which the diffusion-driven instability occurs. Find the critical wavenumber at the bifurcation to such a diffusion-driven instability.
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Paper 2, Section II, 20G
Part II, 2018 commentLet be a prime, and let . Let .
(a) Show that .
(b) Calculate . Deduce that .
(c) Now suppose . Prove that . [You may use any general result without proof, provided that you state it precisely.]
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Paper 4, Section II, G
Part II, 2018 commentLet be a square-free integer, and let be an integer. Let .
(a) By considering the factorisation of into prime ideals, show that .
(b) Let be the bilinear form defined by . Let . Calculate the dual basis of with respect to , and deduce that .
(c) Show that if is a prime and , then .
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Paper 1, Section II, G
Part II, 2018 comment(a) Let be an integer such that is prime. Suppose that the ideal class group of is trivial. Show that if is an integer and , then is prime.
(b) Show that the ideal class group of is trivial.
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Paper 1, Section I, G
Part II, 2018 comment(a) State and prove the Chinese remainder theorem.
(b) An integer is squarefull if whenever is prime and , then . Show that there exist 1000 consecutive positive integers, none of which are squarefull.
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Paper 2, Section I, G
Part II, 2018 commentDefine the Legendre symbol, and state Gauss's lemma. Show that if is an odd prime, then
Use the law of quadratic reciprocity to compute .
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Paper 3, Section I, G
Part II, 2018 commentWhat is a multiplicative function? Show that if is a multiplicative function, then so is .
Define the Möbius function , and show that it is multiplicative. Deduce that
and that
What is if What is if
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Paper 4, Section I, G
Part II, 2018 commentShow that if a continued fraction is periodic, then it represents a quadratic irrational. What number is represented by the continued fraction ?
Compute the continued fraction expansion of . Hence or otherwise find a solution in positive integers to the equation .
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Paper 4, Section II, G
Part II, 2018 comment(a) State and prove the Fermat-Euler theorem. Let be a prime and a positive integer. Show that holds for every integer if and only if .
(b) Let be an odd integer and be an integer with . What does it mean to say that is a Fermat pseudoprime to base ? What does it mean to say that is a Carmichael number?
Show that every Carmichael number is squarefree, and that if is squarefree, then is a Carmichael number if and only if for every prime divisor of . Deduce that a Carmichael number is a product of at least three primes.
(c) Let be a fixed odd prime. Show that there are only finitely many pairs of primes for which is a Carmichael number.
[You may assume throughout that is cyclic for every odd prime and every integer
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Paper 3, Section II, G
Part II, 2018 commentWhat does it mean to say that a positive definite binary quadratic form is reduced? What does it mean to say that two binary quadratic forms are equivalent? Show that every positive definite binary quadratic form is equivalent to some reduced form.
Show that the reduced positive definite binary quadratic forms of discriminant are and . Show also that a prime is represented by if and only if
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Paper 4, Section II, E
Part II, 2018 commentThe inverse discrete Fourier transform is given by the formula
Here, is the primitive root of unity of degree and
(a) Show how to assemble in a small number of operations if the Fourier transforms of the even and odd parts of ,
are already known.
(b) Describe the Fast Fourier Transform (FFT) method for evaluating , and draw a diagram to illustrate the method for .
(c) Find the cost of the FFT method for (only multiplications count).
(d) For use the FFT method to find when: (i) , (ii) .
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Paper 2, Section II, E
Part II, 2018 commentThe Poisson equation in the unit interval , with , is discretised with the formula
where , the grid points are at and .
(a) Write the above system of equations in the vector form and describe the relaxed Jacobi method with relaxation parameter for solving this linear system.
(b) For and being the exact and the iterated solution, respectively, let be the error and be the iteration matrix, so that
Express in terms of the matrix and the relaxation parameter . Using the fact that for any Toeplitz symmetric tridiagonal matrix, the eigenvectors have the form:
find the eigenvalues of . Hence deduce the eigenvalues of .
(c) For as above, let
be the expansion of the error with respect to the eigenvectors of .
Find the range of the parameter which provides convergence of the method for any , and prove that, for any such , the rate of convergence is not faster than when is large.
(d) Show that, for an appropriate range of , the high frequency components of the error tend to zero much faster than the rate obtained in part (c). Determine the optimal parameter which provides the largest supression of the high frequency components per iteration, and find the corresponding attenuation factor assuming is large. That is, find the least such that for .
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Paper 1, Section II, E
Part II, 2018 comment(a) Suppose that is a real matrix, and and are given so that . Further, let be a non-singular matrix such that , where is the first coordinate vector and .
Let . Prove that the eigenvalues of are together with the eigenvalues of the bottom right submatrix of .
Explain briefly how, given a vector , an orthogonal matrix such that can be constructed.
(b) Suppose that is a real matrix, and two linearly independent vectors are given such that the linear subspace spanned by and is invariant under the action of , i.e.,
Denote by an matrix whose two columns are the vectors and , and let be a non-singular matrix such that is upper triangular:
Again, let . Prove that the eigenvalues of are the eigenvalues of the top left submatrix of together with the eigenvalues of the bottom right submatrix of .
Explain briefly how, for given vectors , an orthogonal matrix which satisfies (*) can be constructed.
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Paper 3, Section II, E
Part II, 2018 commentThe diffusion equation for :
is solved numerically by the difference scheme
Here is the Courant number, with , and .
(a) Prove that, as with constant , the local error of the method is .
(b) Applying the Fourier stability analysis, show that the method is stable if and only if . [Hint: If a polynomial has real roots, then those roots lie in if and only if and .]
(c) Prove that, for the same equation, the leapfrog scheme
is unstable for any choice of .
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Paper 4, Section II,
Part II, 2018 commentConsider the deterministic system
where and are scalars. Here is the state variable and the control variable is to be chosen to minimise, for a fixed , the cost
where is known and for all . Let be the minimal cost from state and time .
(a) By writing the dynamic programming equation in infinitesimal form and taking the appropriate limit show that satisfies
with boundary condition .
(b) Determine the form of the optimal control in the special case where is constant, and also in general.
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Paper 3, Section II, K
Part II, 2018 commentThe scalars are related by the equations
where the initial state is normally distributed with mean and variance 1 and is a sequence of independent random variables each normally distributed with mean 0 and variance 1 . The control variable is to be chosen at time on the basis of information , where and
(a) Let be the Kalman filter estimates of , i.e.
where is chosen to minimise . Calculate and show that, conditional on is normally distributed with mean and variance .
(b) Define
Show that , where and .
(c) Show that the minimising control can be expressed in the form and find . How would the expression for be altered if or had variances other than 1?
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Paper 2, Section II,
Part II, 2018 comment(a) A ball may be in one of boxes. A search of the box costs and finds the ball with probability if the ball is in that box. We are given initial probabilities that the ball is in the box.
Show that the policy which at time searches the box with the maximal value of minimises the expected searching cost until the ball is found, where is the probability (given everything that has occurred up to time ) that the ball is in box .
(b) Next suppose that a reward is earned if the ball is found in the box. Suppose also that we may decide to stop at any time. Develop the dynamic programming equation for the value function starting from the probability distribution .
Show that if then it is never optimal to stop searching until the ball is found. In this case, is the policy defined in part (a) optimal?
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Paper 4, Section II, D
Part II, 2018 commentThe spin operators obey the commutation relations . Let be an eigenstate of the spin operators and , with and . Show that
where . When , use this to derive the explicit matrix representation
in a basis in which is diagonal.
A beam of atoms, each with spin 1 , is polarised to have spin along the direction . This beam enters a Stern-Gerlach filter that splits the atoms according to their spin along the -axis. Show that , where (respectively, ) is the number of atoms emerging from the filter with spins parallel (respectively, anti-parallel) to .
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Paper 1, Section II, D
Part II, 2018 commentA one-dimensional harmonic oscillator has Hamiltonian
where . Show that , where and .
This oscillator is perturbed by adding a new term to the Hamiltonian. Given that
show that the ground state of the perturbed system is
to first order in . [You may use the fact that, in non-degenerate perturbation theory, a perturbation causes the first-order shift
in the energy level.]
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Paper 3, Section II, D
Part II, 2018 commentA quantum system is prepared in the ground state at time . It is subjected to a time-varying Hamiltonian . Show that, to first order in , the system evolves as
where and
A large number of hydrogen atoms, each in the ground state, are subjected to an electric field
where is a constant. Show that the fraction of atoms found in the state is, after a long time and to lowest non-trivial order in ,
where is the energy difference between the and states, and is the electron charge and the Bohr radius. What fraction of atoms lie in the state?
[Hint: You may assume the hydrogenic wavefunctions
and the integral
for a positive integer.]
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Paper 2, Section II, D
Part II, 2018 commentExplain what is meant by the intrinsic parity of a particle.
In each of the decay processes below, parity is conserved.
A deuteron has intrinsic parity and spin . A negatively charged pion has spin . The ground state of a hydrogenic 'atom' formed from a deuteron and a pion decays to two identical neutrons , each of spin and parity . Deduce the intrinsic parity of the pion.
The particle has spin and decays as
What are the allowed values of the orbital angular momentum? In the centre of mass frame, the vector joining the pion to the neutron makes an angle to the -axis. The final state is an eigenstate of and the spatial probability distribution is proportional to . Deduce the intrinsic parity of the .
[Hint: You may use the fact that the first three Legendre polynomials are given by
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Paper 4, Section II,
Part II, 2018 commentLet be an unknown function, twice continuously differentiable with for all . For some , we know the value and we wish to estimate its derivative . To do so, we have access to a pseudo-random number generator that gives i.i.d. uniform over , and a machine that takes input and returns , where the are i.i.d. .
(a) Explain how this setup allows us to generate independent , where the take value 1 or with probability , for any .
(b) We denote by the output . Show that for some independent
(c) Using the intuition given by the least-squares estimator, justify the use of the estimator given by
(d) Show that
Show that for some choice of parameter , this implies
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Paper 3, Section II, K
Part II, 2018 commentIn the model of a Gaussian distribution in dimension , with unknown mean and known identity covariance matrix , we estimate based on a sample of i.i.d. observations drawn from .
(a) Define the Fisher information , and compute it in this model.
(b) We recall that the observed Fisher information is given by
Find the limit of , where is the maximum likelihood estimator of in this model.
(c) Define the Wald statistic and compute it. Give the limiting distribution of and explain how it can be used to design a confidence interval for .
[You may use results from the course provided that you state them clearly.]
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Paper 2, Section II,
Part II, 2018 commentWe consider the model of a Gaussian distribution in dimension , with unknown mean and known identity covariance matrix . We estimate based on one observation , under the loss function
(a) Define the risk of an estimator . Compute the maximum likelihood estimator of and its risk for any .
(b) Define what an admissible estimator is. Is admissible?
(c) For any , let be the prior . Find a Bayes optimal estimator under this prior with the quadratic loss, and compute its Bayes risk.
(d) Show that is minimax.
[You may use results from the course provided that you state them clearly.]
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Paper 1, Section II,
Part II, 2018 commentA scientist wishes to estimate the proportion of presence of a gene in a population of flies of size . Every fly receives a chromosome from each of its two parents, each carrying the gene with probability or the gene with probability , independently. The scientist can observe if each fly has two copies of the gene A (denoted by AA), two copies of the gene (denoted by BB) or one of each (denoted by AB). We let , and denote the number of each observation among the flies.
(a) Give the probability of each observation as a function of , denoted by , for all three values , or .
(b) For a vector , we let denote the estimator defined by
Find the unique vector such that is unbiased. Show that is a consistent estimator of .
(c) Compute the maximum likelihood estimator of in this model, denoted by . Find the limiting distribution of . [You may use results from the course, provided that you state them clearly.]
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Paper 4, Section II, J
Part II, 2018 commentLet be a measurable space. Let be a measurable map, and a probability measure on .
(a) State the definition of the following properties of the system :
(i) is T-invariant.
(ii) is ergodic with respect to .
(b) State the pointwise ergodic theorem.
(c) Give an example of a probability measure preserving system in which for -a.e. .
(d) Assume is finite and is the boolean algebra of all subsets of . Suppose that is a -invariant probability measure on such that for all . Show that is a bijection.
(e) Let , the set of positive integers, and be the -algebra of all subsets of . Suppose that is a -invariant ergodic probability measure on . Show that there is a finite subset with .
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Paper 2, Section II, J
Part II, 2018 commentLet be a probability space. Let be a sequence of random variables with for all .
(a) Suppose is another random variable such that . Why is integrable for each ?
(b) Assume for every random variable on such that . Show that there is a subsequence , such that
(c) Assume that in probability. Show that . Show that in . Must it converge also in Justify your answer.
(d) Assume that the are independent. Give a necessary and sufficient condition on the sequence for the sequence
to converge in .
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Paper 3, Section II, J
Part II, 2018 commentLet be the Lebesgue measure on the real line. Recall that if is a Borel subset, then
where the infimum is taken over all covers of by countably many intervals, and denotes the length of an interval .
(a) State the definition of a Borel subset of .
(b) State a definition of a Lebesgue measurable subset of .
(c) Explain why the following sets are Borel and compute their Lebesgue measure:
(d) State the definition of a Borel measurable function .
(e) Let be a Borel measurable function . Is it true that the subset of all where is continuous at is a Borel subset? Justify your answer.
(f) Let be a Borel subset with . Show that
contains the interval .
(g) Let be a Borel subset such that . Show that for every , there exists in such that
Deduce that contains an open interval around 0 .
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Paper 1, Section II, J
Part II, 2018 comment(a) Let be a real random variable with . Show that the variance of is equal to .
(b) Let be the indicator function of the interval on the real line. Compute the Fourier transform of .
(c) Show that
(d) Let be a real random variable and be its characteristic function.
(i) Assume that for some . Show that there exists such that almost surely:
(ii) Assume that for some real numbers , not equal to 0 and such that is irrational. Prove that is almost surely constant. [Hint: You may wish to consider an independent copy of .]
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Paper 4, Section I, 10D
Part II, 2018 commentLet denote the set of all -bit strings. Suppose we are given a 2-qubit quantum gate which is promised to be of the form
but the 2-bit string is unknown to us. We wish to determine with the least number of queries to . Define , where is the identity operator and .
(a) Is unitary? Justify your answer.
(b) Compute the action of on , and the action of on , in each case expressing your answer in terms of and . Hence or otherwise show that may be determined with certainty using only one application of the gate , together with any other gates that are independent of .
(c) Let be the function having value 0 for all and having value 1 for . It is known that a single use of can be implemented with a single query to a quantum oracle for the function . But suppose instead that we have a classical oracle for , i.e. a black box which, on input , outputs the value of . Can we determine with certainty using a single query to the classical oracle? Justify your answer.
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Paper 3, Section I, 10D
Part II, 2018 commentLet denote the set of all -bit strings. For any Boolean function on 2 bits consider the linear operation on 3 qubits defined by
for all and denoting addition of bits modulo 2 . Here the first register is a 2-qubit register and the second is a 1-qubit register. We are able to apply only the 1-qubit Pauli and Hadamard gates to any desired qubits, as well as the 3 -qubit gate to any three qubits. We can also perform measurements in the computational basis.
(a) Describe how we can construct the state
starting from the standard 3-qubit state .
(b) Suppose now that the gate is given to us but is not specified. However is promised to be one of two following cases:
(i) is a constant function (i.e. for all , or for all ),
(ii) for any 2-bit string we have (with as above).
Show how we may determine with certainty which of the two cases (i) or (ii) applies, using only a single application of .
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Paper 2, Section I,
Part II, 2018 comment(a) The classical controlled- operation applied to the 2-bit string (for or 1 ) achieves the cloning of , i.e. the result is . Let denote the quantum controlled (or controlled-NOT) operation on two qubits. For which qubit states will the application of to (with the first qubit being the control qubit) achieve the cloning of ? Justify your answer.
(b) Let and be two distinct non-orthogonal quantum states. State and prove the quantum no-cloning theorem for unitary processes.
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Paper 1, Section I, D
Part II, 2018 comment(a) Define what it means for a 2-qubit state of a composite quantum system to be entangled.
Consider the 2-qubit state
where is the Hadamard gate. From the definition of entanglement, show that is an entangled state.
(b) Alice and Bob are distantly separated in space. Initially they each hold one qubit of the 2-qubit entangled state
They are able to perform local quantum operations (unitary gates and measurements) on quantum systems they hold. Alice wants to communicate two classical bits of information to Bob. Explain how she can achieve this (within their restricted operational resources) by sending him a single qubit.
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Paper 2, Section II, D
Part II, 2018 comment(a) Suppose that Alice and Bob are distantly separated in space and each has one qubit of the 2-qubit state . They also have the ability to perform local unitary quantum operations and local computational basis measurements, and to communicate only classically. Alice has a 1-qubit state (whose identity is unknown to her) which she wants to communicate to Bob. Show how this can be achieved using only the operational resources, listed above, that they have available.
Suppose now that a third party, called Charlie, joins Alice and Bob. They are all mutually distantly separated in space and each holds one qubit of the 3-qubit state
As previously with Alice and Bob, they are able to communicate with each other only classically, e.g. by telephone, and they can each also perform only local unitary operations and local computational basis measurements. Alice and Bob phone Charlie to say that they want to do some quantum teleportation and they need a shared state (as defined above). Show how Charlie can grant them their wish (with certainty), given their joint possession of and using only their allowed operational resources. [Hint: It may be useful to consider application of an appropriate Hadamard gate action.]
(b) State the quantum no-signalling principle for a bipartite state of the composite system .
Suppose we are given an unknown one of the two states
and we wish to identify which state we have. Show that the minimum error probability for this state discrimination task is zero.
Suppose now that we have access only to qubit of the received state. Show that we can now do no better in the state discrimination task than just making a random guess as to which state we have.
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Paper 3, Section II,
Part II, 2018 commentIn this question you may assume the following fact about the quantum Fourier transform if and , where , then
where .
(a) Let denote the integers modulo . Let be a periodic function with period and with the property that is one-to-one within each period. We have one instance of the quantum state
and the ability to calculate the function on at most two values of our choice.
Describe a procedure that may be used to determine the period with success probability . As a further requirement, at the end of the procedure we should know if it has been successful, or not, in outputting the correct period value. [You may assume that the number of integers less than that are coprime to is .
(b) Consider the function defined by .
(i) Show that is periodic and find the period.
(ii) Suppose we are given the state and we measure the second register. What are the possible resulting measurement values and their probabilities?
(iii) Suppose the measurement result was . Find the resulting state of the first register after the measurement.
(iv) Suppose we measure the state (with from part (iii)). What is the probability of each outcome ?
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Paper 1, Section II, I
Part II, 2018 comment(a) Define the derived subgroup, , of a finite group . Show that if is a linear character of , then ker . Prove that the linear characters of are precisely the lifts to of the irreducible characters of . [You should state clearly any additional results that you require.]
(b) For , you may take as given that the group
has order .
(i) Let . Show that if is any -th root of unity in , then there is a representation of over which sends
(ii) Find all the irreducible representations of .
(iii) Find the character table of .
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Paper 2, Section II, I
Part II, 2018 comment(a) Suppose is a subgroup of a finite group is an irreducible character of and are the irreducible characters of . Show that in the restriction , the multiplicities satisfy
Determine necessary and sufficient conditions under which the inequality in ( ) is actually an equality.
(b) Henceforth suppose that is a (normal) subgroup of index 2 in , and that is an irreducible character of .
Lift the non-trivial linear character of to obtain a linear character of which satisfies
(i) Show that the following are equivalent:
(1) is irreducible;
(2) for some with ;
(3) the characters and of are not equal.
(ii) Suppose now that is irreducible. Show that if is an irreducible character of which satisfies
then either or
(iii) Suppose that is the sum of two irreducible characters of , say . If is an irreducible character of such that has or as a constituent, show that .
(c) Suppose that is a finite group with a subgroup of index 3 , and let be an irreducible character of . Prove that
Give examples to show that each possibility can occur, giving brief justification in each case.
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Paper 3, Section II, I
Part II, 2018 commentState the row orthogonality relations. Prove that if is an irreducible character of the finite group , then divides the order of .
Stating clearly any additional results you use, deduce the following statements:
(i) Groups of order , where is prime, are abelian.
(ii) If is a group of order , where is prime, then either the degrees of the irreducible characters of are all 1 , or they are
(iii) No simple group has an irreducible character of degree 2 .
(iv) Let and be prime numbers with , and let be a non-abelian group of order . Then divides and has conjugacy classes.
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Paper 4, Section II, I
Part II, 2018 commentDefine and write down a complete list
of its continuous finite-dimensional irreducible representations. You should define all the terms you use but proofs are not required. Find the character of . State the Clebsch-Gordan formula.
(a) Stating clearly any properties of symmetric powers that you need, decompose the following spaces into irreducible representations of :
(i) ;
(ii) (with multiplicands);
(iii) .
(b) Let act on the space of complex matrices by
where is the block matrix . Show that this gives a representation of and decompose it into irreducible summands.
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Paper 2, Section II, F
Part II, 2018 commentState the uniformisation theorem. List without proof the Riemann surfaces which are uniformised by and those uniformised by .
Let be a domain in whose complement consists of more than one point. Deduce that is uniformised by the open unit disk.
Let be a compact Riemann surface of genus and be distinct points of . Show that is uniformised by the open unit disk if and only if , and by if and only if or .
Let be a lattice and a complex torus. Show that an analytic map is either surjective or constant.
Give with proof an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.
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Paper 3, Section II, F
Part II, 2018 commentDefine the degree of an analytic map of compact Riemann surfaces, and state the Riemann-Hurwitz formula.
Let be a lattice in and the associated complex torus. Show that the
is biholomorphic with four fixed points in .
Let be the quotient surface (the topological surface obtained by identifying and ), and let be the associated projection map. Denote by the complement of the four fixed points of , and let . Describe briefly a family of charts making into a Riemann surface, so that is a holomorphic map.
Now assume that, by adding finitely many points, it is possible to compactify to a Riemann surface so that extends to a regular map . Find the genus of .
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Paper 1, Section II, F
Part II, 2018 commentGiven a complete analytic function on a domain , define the germ of a function element of at . Let be the set of all germs of function elements in . Describe without proofs the topology and complex structure on and the natural covering map . Prove that the evaluation map defined by
is analytic on each component of .
Suppose is an analytic map of compact Riemann surfaces with the set of branch points. Show that is a regular covering map.
Given , explain how any closed curve in with initial and final points yields a permutation of the set . Show that the group obtained from all such closed curves is a transitive subgroup of the group of permutations of .
Find the group for the analytic map where .
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Paper 4, Section I, J
Part II, 2018 commentA scientist is studying the effects of a drug on the weight of mice. Forty mice are divided into two groups, control and treatment. The mice in the treatment group are given the drug, and those in the control group are given water instead. The mice are kept in 8 different cages. The weight of each mouse is monitored for 10 days, and the results of the experiment are recorded in the data frame Weight.data. Consider the following code and its output.
head (Weight.data)
Time Group Cage Mouse Weight
11 Control 1 1
Control 1 1
Control
44 Control
Control 1 1
Control
(Weight Time*Group Cage, data=Weight. data)
Call:
(formula Weight Time Group Cage, data Weight. data)
Residuals:
Min Median Max
Coefficients:
Estimate Std. Error t value
GroupTreatment
Cage2
Time: GroupTreatment
Signif. codes: 0 '' '' '' '., ', 1
Residual standard error: on 391 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 8 and 391 DF, p-value:
Which parameters describe the rate of weight loss with time in each group? According to the output, is there a statistically significant weight loss with time in the control group?
Three diagnostic plots were generated using the following code.

Weight.data$Time[mouse1]

Weight.data$Time[mouse2]

Based on these plots, should you trust the significance tests shown in the output of the command summary (mod1)? Explain.
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Paper 3, Section I, J
Part II, 2018 commentThe data frame Cases. of .flu contains a list of cases of flu recorded in 3 London hospitals during each month of 2017 . Consider the following code and its output.
table (Cases. of.flu)
Month Hospital
May
November
October
September
Cases. of.flu.table = as.data.frame (table (Cases. of .flu))
head (Cases. of .flu.table)
Month Hospital Freq
1 April A 10
2 August A 9
3 December A 24
4 February A 49
5 January A 45
6 July A 5
glm (Freq ., data=Cases. of .flu.table, family=poisson)
[1]
levels (Cases. of.flu$Month)

Describe a test for the null hypothesis of independence between the variables Month and Hospital using the deviance statistic. State the assumptions of the test.
Perform the test at the level for each of the two different models shown above. You may use the table below showing 99 th percentiles of the distribution with a range of degrees of freedom . How would you explain the discrepancy between their conclusions?

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Paper 2, Section I,
Part II, 2018 commentConsider a linear model with , where the design matrix is by . Provide an expression for the -statistic used to test the hypothesis for . Show that it is a monotone function of a log-likelihood ratio statistic.
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Paper 1, Section I, J
Part II, 2018 commentThe data frame Ambulance contains data on the number of ambulance requests from a Cambridgeshire hospital on different days. In addition to the number of ambulance requests on each day, the dataset records whether each day fell in the winter season, on a weekend, or on a bank holiday, as well as the pollution level on each day.
A health researcher fitted two models to the dataset above using . Consider the following code and its output.




Define the two models fitted by this code and perform a hypothesis test with level in which one of the models is the null hypothesis and the other is the alternative. State the theorem used in this hypothesis test. You may use the information generated by the following commands.

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Paper 4, Section II, J
Part II, 2018 commentBridge is a card game played by 2 teams of 2 players each. A bridge club records the outcomes of many games between teams formed by its members. The outcomes are modelled by
where is a parameter representing the skill of player , and is a parameter representing how well-matched the team formed by and is.
(a) Would it make sense to include an intercept in this logistic regression model? Explain your answer.
(b) Suppose that players 1 and 2 always play together as a team. Is there a unique maximum likelihood estimate for the parameters and ? Explain your answer.
(c) Under the model defined above, derive the asymptotic distribution (including the values of all relevant parameters) for the maximum likelihood estimate of the probability that team wins a game against team . You can state it as a function of the true vector of parameters , and the Fisher information matrix with games. You may assume that as , and that has a unique maximum likelihood estimate for large enough.
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Paper 1, Section II, J
Part II, 2018 commentA clinical study follows a number of patients with an illness. Let be the length of time that patient lives and a vector of predictors, for . We shall assume that are independent. Let and be the probability density function and cumulative distribution function, respectively, of . The hazard function is defined as
We shall assume that , where is a vector of coefficients and is some fixed hazard function.
(a) Prove that .
(b) Using the equation in part (a), write the log-likelihood function for in terms of and only.
(c) Show that the maximum likelihood estimate of can be obtained through a surrogate Poisson generalised linear model with an offset.
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Paper 4, Section II, A
Part II, 2018 commentThe one-dimensional Ising model consists of a set of spins with Hamiltonian
where periodic boundary conditions are imposed so . Here is a positive coupling constant and is an external magnetic field. Define a matrix with elements
where indices take values and with Boltzmann's constant and temperature.
(a) Prove that the partition function of the Ising model can be written as
Calculate the eigenvalues of and hence determine the free energy in the thermodynamic limit . Explain why the Ising model does not exhibit a phase transition in one dimension.
(b) Consider the case of zero magnetic field . The correlation function is defined by
(i) Show that, for ,
(ii) By diagonalizing , or otherwise, calculate for any positive integer . Hence show that
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Paper 1, Section II, A
Part II, 2018 comment(a) A macroscopic system has volume and contains particles. Let denote the number of states of the system which have energy in the range where represents experimental uncertainty. Define the entropy of the system and explain why the dependence of on is usually negligible. Define the temperature and pressure of the system and hence obtain the fundamental thermodynamic relation.
(b) A one-dimensional model of rubber consists of a chain of links, each of length a. The chain lies along the -axis with one end fixed at and the other at where . The chain can "fold back" on itself so may not increase monotonically along the chain. Let and denote the number of links along which increases and decreases, respectively. All links have the same energy.
(i) Show that and are uniquely determined by and . Determine , the number of different arrangements of the chain, as a function of and . Hence show that, if and then the entropy of the chain is
where is Boltzmann's constant. [You may use Stirling's approximation: ! for
(ii) Let denote the force required to hold the end of the chain fixed at . This force does work on the chain if the length is increased by . Write down the fundamental thermodynamic relation for this system and hence calculate as a function of and the temperature .
Assume that . Show that the chain satisfies Hooke's law . What happens if is held constant and is increased?
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Paper 3, Section II, A
Part II, 2018 comment(a) A system of non-interacting bosons has single particle states with energies . Show that the grand canonical partition function is
where is Boltzmann's constant, and is the chemical potential. What is the maximum possible value for ?
(b) A system of bosons has one energy level with zero energy and energy levels with energy . The number of particles with energies is respectively.
(i) Write down expressions for and in terms of and .
(ii) At temperature what is the maximum possible number of bosons in the state with energy What happens for
(iii) Calculate the temperature at which Bose condensation occurs.
(iv) For , show that . For show that
(v) Calculate the mean energy for and for . Hence show that the heat capacity of the system is
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Paper 2, Section II, A
Part II, 2018 comment(a) Starting from the canonical ensemble, derive the Maxwell-Boltzmann distribution for the velocities of particles in a classical gas of atoms of mass . Derive also the distribution of speeds of the particles. Calculate the most probable speed.
(b) A certain atom emits photons of frequency . A gas of these atoms is contained in a box. A small hole is cut in a wall of the box so that photons can escape in the positive -direction where they are received by a detector. The frequency of the photons received is Doppler shifted according to the formula
where is the -component of the velocity of the atom that emits the photon and is the speed of light. Let be the temperature of the gas.
(i) Calculate the mean value of .
(ii) Calculate the standard deviation .
(iii) Show that the relative number of photons received with frequency between and is where
for some coefficient to be determined. Hence explain how observations of the radiation emitted by the gas can be used to measure its temperature.
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Paper 4, Section II, K
Part II, 2018 commentConsider a utility function , which is assumed to be concave, strictly increasing and twice differentiable. Further, satisfies
for some positive constants and . Let be an -distributed random variable and set .
(a) Show that
(b) Show that and . Discuss this result in the context of meanvariance analysis.
(c) Show that is concave in and , i.e. check that the matrix of second derivatives is negative semi-definite. [You may use without proof the fact that if a matrix has nonpositive diagonal entries and a non-negative determinant, then it is negative semi-definite.]
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Paper 3, Section II,
Part II, 2018 commentConsider a multi-period model with asset prices , modelled on a probability space and adapted to a filtration . Assume that is -trivial, i.e. for all , and assume that is a -a.s. strictly positive numéraire, i.e. -a.s. for all . Further, let denote the discounted price process defined by .
(a) What does it mean to say that a self-financing strategy is an arbitrage?
(b) State the fundamental theorem of asset pricing.
(c) Let be a probability measure on which is equivalent to and for which for all . Show that the following are equivalent:
(i) is a martingale measure.
(ii) If is self-financing and is bounded, i.e. for a suitable , then the value process of is a -martingale.
(iii) If is self-financing and is bounded, then the value process of satisfies
[Hint: To show that (iii) implies (i) you might find it useful to consider self-financing strategies with of the form
for any and any .]
(d) Prove that if there exists a martingale measure satisfying the conditions in (c) then there is no arbitrage.
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Paper 2, Section II, K
Part II, 2018 commentConsider the Black-Scholes model, i.e. a market model with one risky asset with price at time given by
where denotes a Brownian motion on the constant growth rate, the constant volatility and the initial price of the asset. Assume that the riskless rate of interest is .
(a) Consider a European option with expiry for any bounded, continuous function . Use the Cameron-Martin theorem to characterize the equivalent martingale measure and deduce the following formula for the price of at time 0 :
(b) Find the price at time 0 of a European option with maturity and payoff for some . What is the value of the option at any time Determine a hedging strategy (you only need to specify how many units of the risky asset are held at any time ).
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Paper 1, Section II, K
Part II, 2018 comment(a) What does it mean to say that is a martingale?
(b) Let be independent random variables on with -a.s. and . Further, let
Show that is a martingale with respect to the filtration .
(c) Let be an adapted process with respect to a filtration such that for every . Show that admits a unique decomposition
where is a martingale and is a previsible process with , which can recursively be constructed from as follows,
(d) Let be a super-martingale. Show that the following are equivalent:
(i) is a martingale.
(ii) for all .
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Paper 1, Section I,
Part II, 2018 commentState and prove Sperner's lemma concerning colourings of points in a triangular grid.
Suppose that is a non-degenerate closed triangle with closed edges and . Show that we cannot find closed sets with , for , such that
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Paper 2, Section I,
Part II, 2018 commentFor we write . Define
(a) Suppose that is a convex subset of , that and that for all . Show that for all .
(b) Suppose that is a non-empty closed bounded convex subset of . Show that there is a such that for all . If for each with , show that
for all , and that is unique.
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Paper 3, Section I,
Part II, 2018 commentState a version of the Baire category theorem and use it to prove the following result:
If is analytic, but not a polynomial, then there exists a point such that each coefficient of the Taylor series of at is non-zero.
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Paper 4, Section I,
Part II, 2018 commentLet and . If we have an infinite sequence of integers with , show that
is irrational.
Does the result remain true if the are not restricted to integer values? Justify your answer.
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Paper 2, Section II, F
Part II, 2018 comment(a) Give Bernstein's probabilistic proof of Weierstrass's theorem.
(b) Are the following statements true or false? Justify your answer in each case.
(i) If is continuous, then there exists a sequence of polynomials converging pointwise to on .
(ii) If is continuous, then there exists a sequence of polynomials converging uniformly to on .
(iii) If is continuous and bounded, then there exists a sequence of polynomials converging uniformly to on .
(iv) If is continuous and are distinct points in , then there exists a sequence of polynomials with , for , converging uniformly to on .
(v) If is times continuously differentiable, then there exists a sequence of polynomials such that uniformly on for each .
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Paper 4, Section II, F
Part II, 2018 commentWe work in . Consider
and
Show that if is analytic, then there is a sequence of polynomials such that uniformly on .
Show that there is a sequence of polynomials such that uniformly for and uniformly for .
Give two disjoint non-empty bounded closed sets and such that there does not exist a sequence of polynomials with uniformly on and uniformly on . Justify your answer.
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Paper 4, Section II, C
Part II, 2018 commentA physical system permits one-dimensional wave propagation in the -direction according to the equation
Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wavenumber. Waves of what wavenumber are at the front of a dispersing wave train arising from a localised initial disturbance? For waves of what wavenumbers do wave crests move faster or slower than a packet of waves?
Find the solution of the above equation for the initial disturbance given by
where , and is the complex conjugate of . Let be held fixed. Use the method of stationary phase to obtain a leading-order approximation to this solution for large when , where the solutions for the stationary points should be left in implicit form.
Very briefly discuss the nature of the solutions for and .
[Hint: You may quote the result that the large time behaviour of
due to a stationary point , is given by
where
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Paper 2, Section II, C
Part II, 2018 commentA perfect gas occupies the region of a tube that lies parallel to the -axis. The gas is initially at rest, with density , pressure , speed of sound and specific heat ratio . For times a piston, initially at , is pushed into the gas at a constant speed . A shock wave propagates at constant speed into the undisturbed gas ahead of the piston. Show that the excess pressure in the gas next to the piston, , is given implicitly by the expression
Show also that
and interpret this result.
[Hint: You may assume for a perfect gas that the speed of sound is given by
and that the internal energy per unit mass is given by
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Paper 1, Section II, 39C
Part II, 2018 commentDerive the wave equation governing the velocity potential for linearised sound waves in a perfect gas. How is the pressure disturbance related to the velocity potential?
A high pressure gas with unperturbed density is contained within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Let the metal shell have radius , mass per unit surface area, and an elastic stiffness which tries to restore the radius to its equilibrium value with a force per unit surface area. Assume that there is a vacuum outside the spherical shell. Show that the frequencies of vibration satisfy
where , and is the speed of sound in the undisturbed gas. Briefly comment on the existence of solutions.
[Hint: In terms of spherical polar coordinates you may assume that for a function ,
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Paper 3, Section II, 40C
Part II, 2018 commentDerive the ray-tracing equations
for wave propagation through a slowly-varying medium with local dispersion relation , where and are the frequency and wavevector respectively, is time and are spatial coordinates. The meaning of the notation should be carefully explained.
A slowly-varying medium has a dispersion relation , where . State and prove Snell's law relating the angle between a ray and the -axis to .
Consider the case of a medium with wavespeed , where and are positive constants. Show that a ray that passes through the origin with wavevector , remains in the region
By considering an approximation to the equation for a ray in the region , or otherwise, determine the path of a ray near , and hence sketch rays passing through the origin for a few sample values of in the range .